เลขยกกำลัง

แบบฝึกทักษะคณิตศาสตร์

กลุ่มสาระการเรยี นร้คู ณิตศาสตร์

1.ชนั้ มัธยมศกึ ษาปที ี่

--------------->

เลขยกกำลัง

Exponent

โรงเรยี นบ้านมอเจรญิ อาเภอปางศลิ าทอง จังหวัดกาแพงเพชร

สานักงานเขตพืน้ ทีก่ ารศกึ ษาประถมศึกษากาแพงเพชร เขต 2

กระทรวงศกึ ษาธิการ

1

3.1 ความหมายของเลขยกกำลัง

บทนิยาม : ถ้า a แทนจำนวนใด ๆ และ n เปน็ จำนวนเตม็ บวก
“ a ยกกำลงั n “ เขยี นแทนด้วย an มีความหมายดงั น้ี
an = a x a x a x ….. x a
n ตวั
เรียก an วา่ เลขยกกำลัง มี a เปน็ ฐาน และ n เปน็ เลขชกี้ ำลงั

 2  4
 3 
ตวั อย่าง จงหาค่าของ 35 และ

วธิ ีทำ 1.) 35 = 3x3x3x3x3

= 243

ตอบ 35 = 243

2.)  2 4 = 2 x 2 x 2 x 2
 3  3 3 3 3

= 16
81
4
ตอบ  2  = 16
 3 81

2

แบบฝกึ ทักษะที่ 1

ความหมายของเลขยกกำลงั

1. จงเขยี นจำนวนตอ่ ไปนี้ ใหอ้ ยู่ในรูปของการคูณ

1.) 53 = ………………………………………………………………………..

2.) (-1)5 = ………………………………………………………………………..

3.)  2 5 = ………………………………………………………………………..
 3 

4.) (0.7)3 = ………………………………………………………………………..

5.) 125 = ………………………………………………………………………..

6.) (-2.6)4 = ………………………………………………………………………..

7.)  − 1 5 = ……………………………………………………………….
 10 

8.) 6252 = ………………………………………………………………………..

9.) ( a + b )4= ………………………………………………………………

10.)  x 5 = ………………………………………………………………………..
 y 

11.)(-3)5 = ………………………………………………………………………..

12.)  1 3 = ………………………………………………………………………..
 2 

13.)(0.2)2 = ………………………………………………………………………..

14.)(-2)2 = ………………………………………………………………………..

15.)(-10)3 = ………………………………………………………………………..

16.)106 = ………………………………………………………………………..

17.)( 2 x 3 )3= ………………………………………………………………

18.)  5 4 = ………………………………………………………………………..
 3 

19.)( a x b )2= ………………………………………………………………

20.)(2t)7 = ………………………………………………………………………..

3

2. จงเขียนจำนวนต่อไปนี้ให้อยู่ในรูปเลขยกกำลัง

2.1) 2 x 2 x 2 x 2 x 2 x 2 = …………………

2.2) (-4) x (-4) x (-4) x (-4) = …………………

2.3) (0.2) x (0.2) x (0.2) = …………………

2.4)  − 2  x − 2  x − 2  x − 2  x − 2  = …………………
 3  3  3  3  3  = …………………

2.5) 78 x 78 x 78 x 78 x 78 x 78 x 78

2.6) B x B x B x B x …….. x B ( 100 ตัว ) = …………………

2.7) (-1.9) x (-1.9) x (-1.9) x ……… x (-1.9) ( m ตวั ) = …………………

2.8) (-3.4) x (-3.4) x (-3.4) x ………. x (-3.4) ( a + b ตวั ) = …………………

2.9)  10  x  10  x  10  x ………. X  10  ( 19 ตวั ) = …………………
 9   9   9   9  = …………………

2.10) a x a x a x ………. x a ( a ตวั )

2.11) b x b x b x ………. x b ( b ตวั ) = …………………

2.12) ( x+y ) ( x+y ) ( x+y ) ……. ( x+y ) ( x+y ตัว ) = …………………

2.13) (2x-y) (2x-y) (2x-y) (2x-y) = …………………

2.14) (xy) (xy) (xy) ….. (xy) ( m + n ตวั ) = …………………

2.15)  5  x  5  x  5  x  5  x  5  x  5  x  5  = …………………
 7   7   7   7   7   7   7 

2.16)  − 4  x  − 4  x  − 4  x  − 4  x  − 4  x  − 4  = …………………
 5   5   5   5   5   5 

2.17) (0.256)(0.256)(0.256)(0.256) = …………………

2.18)  1  x  1  x  1  x …… x  1  ( x ตวั ) = …………………
 3   3  3   3 
( m ตวั ) = …………………
2.19)  7  x  7  x  7  x ….. x  7  ( 100 ตวั ) = …………………
 5   5   5   5 

2.20) ( MN )( MN )( MN ) ……. ( MN )

4

3. จงเขยี นจำนวนท่ีกำหนดใหต้ ่อไปน้ีในรูปเลขยกกำลัง
ตวั อยา่ ง 16 = 4 x 4 หรือ = 2 x 2 x 2 x 2
= 42 = 24

3.1) 64 = …………………………………………………………………..
= …………………………………………………………………..
3.2) 625 = …………………………………………………………………..
= …………………………………………………………………..
3.3) 81 = …………………………………………………………………..
= …………………………………………………………………..
3.4) 0.125 = …………………………………………………………………..
= …………………………………………………………………..
3.5) 0.0625 = …………………………………………………………………..
= …………………………………………………………………..
3.6) -8 = …………………………………………………………………..
= …………………………………………………………………..
3.7) -27 = …………………………………………………………………..
= …………………………………………………………………..
3.8) 1 = …………………………………………………………………..
16 = …………………………………………………………………..
= …………………………………………………………………..
3.9) 1 = …………………………………………………………………..
− 64 = …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..

= …………………………………………………………………..

= …………………………………………………………………..
= …………………………………………………………………..

= …………………………………………………………………..

= …………………………………………………………………..
= …………………………………………………………………..

5

3.10) 27 = …………………………………………………………………..
3.11) -81 = …………………………………………………………………..
3.12) 196 = …………………………………………………………………..
3.13) 400 = …………………………………………………………………..
3.14) 216 = …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..

3.15) -125 = …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..

3.16) -1,000 = …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..

3.17) 1,024 = …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..

3.18) 2,401 = …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..

3.19) -343 = …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..

3.20) 40 = …………………………………………………………………..
= …………………………………………………………………..
= …………………………………………………………………..

6

3.2 การเขยี นแสดงจำนวนในรปู สัญกรณว์ ิทยาศาสตร์

การเขยี นจำนวนท่ีมีค่าน้อย ๆ หรอื มาก ๆ สามารถเขียนในรูปสญั กรณ์วทิ ยาศาสตร์

บทนิยาม : A x 10n เมอ่ื 1  A  10 และ n เป็นจำนวนเต็ม

ตวั อยา่ งท่ี 1 จงเขยี น 32,000 ในรปู สญั กรณว์ ทิ ยาศาสตร์

วธิ ที ำ 32,000 = 32 x 1,000

= 32 x 103

= 3.2 x 10 x 103

= 3.2 x 104

ตอบ 32,000 = 3.2 x 104

ตวั อยา่ งท่ี 2 จงเขียน 0.0032 ในรูปสัญกรณ์วทิ ยาศาสตร์

วธิ ีทำ 0.0032 = 32
10,000
32
= 104

= 32 x 10-3

= 3.2 x 10 x 10-4

= 3.2 x 101 + (-4)

= 3.2 x 10-3

ตอบ 0.0032 = 3.2 x 10-3

ตัวอย่างท่ี 3 จงหาผลลัพธข์ อง ( 0.04 x 102 ) ( 0.2 x 10-5 )
วิธีทำ ( 0.04 x 102 ) ( 0.2 x 10-5 ) = ( 0.04 x 0.2 ) ( 102 x 10-5 )
= ( 0.008 ) ( 102 + (-5) )
= 8 x 10-3 x 10-3
= 8 x 10(-3) + (-3)
= 8 x 10-6
ตอบ ( 0.04 x 102 ) ( 0.2 x 10-5 ) = 8 x 10-6

7

ตวั อย่างท่ี 4 จงหาผลลัพธ์ของ 246x106
0.02x10−3
246x106 246 106
วธิ ีทำ 0.02x10−3 = 0.02 x 10−3

= 246 x 106
2 x10 −2 10−3
106
= 246 x 10−2 x10 −3
2
106
= 123 x 10−5

= 123 x 106 – (-5)

= 1.23 x 102 x 1011

= 1.23 x 102 + 11

= 1.23 x 1013

ตอบ 246x106 = 1.23 x 1013
0.02x10−3

ตัวอย่างท่ี 5 จงหาผลลัพธ์ของ ( 4.2 x 10-2)2

วธิ ีทำ ( 4.2 x 10-2)2 = ( 4.2 )2 x ( 10-2)2

= 17.64 x 10-4

= 1.764 x 10 x 10-4

= 1.764 x 10-3

ตอบ ( 4.2 x 10-2)2 = 1.764 x 10-3

ตัวอย่างที่ 6 จงหาผลลพั ธ์ของ ( 85 x 104 ) + (65 x 102 )
วธิ ที ำ ( 85 x 104 ) + (65 x 102 ) = ( 85 x 102 x 102 ) + ( 65 x 102 )
= ( 8500 x 102 ) + ( 65 x 102 )
= ( 8500 + 65 ) x 102
= 8565 x 102
= 8.565 x 103 x 102
= 8.565 x 105
ตอบ ( 85 x 104 ) + (65 x 102 ) = 8.565 x 105

8

ตวั อยา่ งที่ 6 จงหาผลลพั ธข์ อง ( 12 x 106 ) – ( 95 x 105 ) + (23 x 104 )
วธิ ที ำ ( 12 x 106 ) – ( 95 x 105 ) + ( 23 x 104 )
= ( 12 x 102 x 104 ) – ( 95 x 10 x 104 ) + ( 23 x
104 )
= (1200 – 950 + 23 ) x 104
= 273 x 104
= 2.73 x 102 x 104
= 2.73 x 106
ตอบ ( 12 x 106 ) – ( 95 x 105 ) + (23 x 104 )= 2.73 x 106

แบบฝึกทกั ษะ 2

การเขียนแสดงจำนวนในรูปสญั กรณว์ ทิ ยาศาสตร์

1. จงเขยี นจำนวนท่กี ำหนดใหใ้ นรูป A x10n เม่ือ 1  A  10 และ n เปน็ จำนวนเตม็

1.1) 450,000 1.2) 3,850,000
= ........................................................... = ...........................................................
= ........................................................... = ...........................................................
1.3) 60,000,000 1.4) 24,560,000
= ........................................................... = ...........................................................
= ........................................................... = ...........................................................
1.5) 0.000078 1.6) 0.000000118
= ........................................................... = ...........................................................
= ........................................................... = ...........................................................
1.7) 0.0001456 1.8) 0.00000286
= ........................................................... = ...........................................................
= ........................................................... = ...........................................................
1.9) 999 x 104 1.10) 5,000 x 10-3
= ........................................................... = ...........................................................
= ........................................................... = ...........................................................
1.11) 71.2 x 10-3 1.12) 399.1 x 10-5
= ........................................................... = ...........................................................
= ........................................................... = ...........................................................

9

1.13) 309 x 10-6 1.14) 62,500 x 10-6
= ........................................................... = ...........................................................
= ........................................................... = ...........................................................
1.15) 79.863 x 10-8 1.16) 0.000047 x 104
= ........................................................... = ...........................................................
= ........................................................... = ...........................................................

2. จงหาผลคูณจากจำนวนที่กำหนดให้ และเขยี นผลลัพธ์อยใู่ นรปู สญั กรณ์วทิ ยาศาสตร์

2.1) ( 2 x 103 ) x ( 3 x 102) 2.2) ( 4 x 103 ) x ( 3 x 103 )

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

2.3) ( 5 x 104 ) x ( 3 x 103 ) 2.4) ( 1.2 x 104 ) x ( 1.2 x 105 )

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

2.5) ( 8 x 107 ) x ( 5 x 106 ) 2.6) ( 4 x 107 ) x ( 2.4 x 108 )

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

2.7) ( 1.8 x 10-2 ) x ( 1.2 x 104 ) 2.8) ( 1.5 x 10-4 ) x (2.2 x 106 )

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

2.9) ( 2.5 x 10-6 ) x ( 2.4 x 10-4 ) 2.10) ( 9.01 x 10-3 ) x (1.5 x 10-7 )

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

2.11) ( 1.2x 10-3 )x( 1.1x 10-5 )x( 3 x 10-6 ) 2.12) ( 2.5 x 10-4 ) x (2.5 x 10-7 ) x ( 4 x 10-5 )

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

10

2.13) (1.1 x 10-5)x(1.2 x 10-6)x(1.3 x 10-7) 2.14) (2.8 x 10-8) x (2.5 x 10-10)x(1 x 10-12)

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

2.15) 16 x10−5 2.16) 428 x103
4 x1012 0.04 x10−5
= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

2.17) 0.016x3x0.0028 2.18) 102 x6x7x0.009
0.0064 0.21x 6000 x10−5
= ...........................................................
= ...........................................................

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

2.19) (80,000,000)2 x (0.000003) 2.20) (4 x102 ) x (5x10−2 )
(0.0002)4 x (600,000)
20 x104
= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

= ........................................................... = ...........................................................

3. จงหาผลลัพธ์อย่ใู นรูปสญั กรณ์วิทยาศาสตร์

3.1) ( 1.22 x 10-2 )2 3.2) ( 0.03 x 104 )3
= ............................................................................ = ..........................................................................
= ............................................................................ = ..........................................................................
= ............................................................................ = ..........................................................................
3.3) ( 2.5 x 10-5 )2 3.4) ( 4.16 x 106 )2
= ............................................................................ = ............................................................................
= ............................................................................ = ............................................................................
= ............................................................................ = ............................................................................
3.5) ( 3.9 x 10-8 )3 3.6) ( 18 x 105 )-3
= ............................................................................ = ............................................................................
= ............................................................................ = ............................................................................
= ............................................................................ = ............................................................................

11

3.7) (5x104) + (2x104) + (7x104) + (4x104) 3.8) (7x105) – (3x105) – (2x105)

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

3.9) (38x106) + (62x105) – (79x105) – (16x105) 3.10) (9x106) – (7x105) – (4x104) – (8x103)

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

3.11) (4x107) + (5x106) – (45x105) – (72x104) 3.12) (35x107) - (9x106) - (243x105) – (572x105)

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

3.13) (8x105 ) + (7x105 ) + (3x105 ) 3.14) (12x1010 ) + (24 x1010 ) + (36 x1010 )
(12x105 ) − (6x105 ) (54 x1010 ) − (36 x1010 )
= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

= ............................................................................ = ............................................................................

4. โมเลกุลของสารชนดิ หนึง่ หนัก 0.000000012 กรมั จงเขียนจำนวนในรูปสญั กรณ์

วิทยาศาสตร์

......................................................................................................................................................

.................................................................................... ..................................................................

.................................................................................................................................... ..................

......................................................................................................................................................

............................................................................................................................. .........................

......................................................................................................................................................

................................................................................................. .....................................................

................................................................................................................................................. .....

12

5. ประเทศไทยมเี นอ้ื ทป่ี ระมาณ 3.236 x 106 ตารางกิโลเมตร จากการสำรวจเมื่อปี 2546
พบว่าประชากรประเทศไทยมีพลเมอื งประมาณ 64,720,000 คน จงหาอัตราส่วนของพ้ืนที่
ต่อคนหน่งึ คน

........................................................................................ ..............................................................
......................................................................................................................................................
......................................................................................................................................................
............................................................................................................................................ ..........
........................................................................................................................ ..............................
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6. ดาวฤกษด์ วงหนง่ึ อย่หู ่างจากโลก 4 ปีแสง ถ้า 1 ปแี สง เท่ากับ 9.05 x 1012 กโิ ลเมตร

ยิงจรวดจากโลกไปยงั ดาวฤกษ์ดวงนี้ดว้ ยความเรว็ 4.42 x 104 กิโลเมตรต่อช่ัวโมง นานก่ี
ช่ัวโมงจรวดจะไปถงึ ดาวฤกษ์ดวงนี้
......................................................................................................................................................
............................................................................................................................. .........................
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7. ในกรงุ เทพฯ มีประชากรประมาณ 5.24 x 107 คน ในปี 2529 อตั ราเกดิ เฉลย่ี แลว้ เป็น
1.5% อยากทราบว่า ในปี พ.ศ. 2529 มเี ด็กเกดิ ใหมป่ ระมาณก่ีคน
......................................................................................................................... .............................
............................................................................................................................. .........................
......................................................................................................................................................
............................................................................................................................. .........................
......................................................................................................................................................
............................................................................................................................. .........................
............................................................................................................................. .........................

13

8. จังหวดั หนึง่ ของประเทศไทยมีพืน้ ทป่ี ระมาณ 4.2 x 109 ตารางเมตร มปี ระชากรอยู่ 3.2 x
10 5 คน จงหาความหนาแน่นของประชากรของจงั หวัดน้ี ตอ่ พน้ื ที่ 1 ตารางเมตร

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9. ไฮโดรเจน 1 ลูกบาศกเ์ ซนตเิ มตรหนักประมาณ 9 x 10-5 กรมั ทองคำ 1 ลกู บาศก์

เซนตเิ มตร หนกั ประมาณ 19.3 กรัม จงหาวา่ ทองคำหนักประมาณก่เี ท่าของไฮโดรเจน
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3.3 การคูณและการหารเลขยกกำลงั

3.3.1 การคณู เลขยกกำลงั ที่มฐี านเดยี วกนั และเลขช้ีกำลังเปน็ จำนวนเต็มบวก

บทนิยาม : ถ้า a แทนจำนวนใด ๆ m และ n เป็นจำนวนเต็มบวก

am x an = am + n

ตัวอย่างท่ี 1 จงหาคา่ ของ 32 x 34

วธิ ที ำ 32 x 34 = ( 3 x 3 ) x ( 3 x 3 x 3 x 3 ) หรือ = 32 + 4
= 36
= 3x3x3x3x3x3

= 36

ตอบ 32 x 34 = 36

14

ตัวอย่างท่ี 2 จงหาคา่ ของ (-2)3 x (-2) หรือ = (-2)3 + 1
วิธีทำ (-2)3 x (-2) = (( -2) x (-2) x (-2)) x (-2) = (-2)4
= (-2) x (-2) x (-2) x (-2)
= (-2)4
ตอบ (-2)3 x (-2) = (-2)4

ตวั อยา่ งท่ี 3 จงหาค่าของ  − 1 2 x  − 1 3
 2   2 

วธิ ีทำ  − 1 2 x  − 1 3 =  − 1  x  − 1  x  − 1  x  − 1  x  − 1 
 2   2   2   2   2   2   2 

=  − 1 5
 2 

 1 2 +3
 2 
หรอื = −

=  − 1 5
 2 

ตอบ  − 1 2 x  − 1 3 =  − 1 5
 2   2   2 

แบบฝกึ ทักษะที่ 3

การคณู เลขยกกำลงั ที่มีฐานเดยี วกันและเลขชก้ี ำลังเป็นจำนวนเตม็ บวก

1. จงเขียนให้อยู่ในรูปอยา่ งง่าย 1.2) A4 x A5
1.1) 83 x 87 = ……………….…………………………
= ……………….…………………………
= ……………….………………………… = ……………….…………………………
= ……………….…………………………
= ……………….………………………… 1.4) (-6)8 x (-6)3
1.3) (1.2)2 x (1.2)6 = ……………….…………………………
= ……………….………………………… = ……………….…………………………
= ……………….………………………… = ……………….…………………………
= ……………….…………………………

15

1.5)  22 5 x  22 8 1.6) ( -5 )7x  −10 3
7 7 2
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.7) a x a2 x a3 1.8) c6 x c5 x c3

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.9)  − 9 16 x  − 9 17 1.10)  3 2 x  3 5
4 4 2 2
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.11)  1 2 x  1 8 x  1 10 1.12) ( 0.3 )19 x  3 10
2 2 2 10
= ……………….…………………………
= ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.13) ( a – b )3 x ( a – b )7 1.14) ( 2a – 1 )6 x ( 2a – 1 )

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.15) E x E2 x E3 x E4 1.16) ( 2y )5 ( 2y )4 ( 2y )3 ( 2y )2
= ……………….…………………………
= ……………….………………………… = ……………….…………………………
= ……………….…………………………
= ……………….…………………………
1.17) (ab)2 x (ab)20 x (ab)200
= ……………….………………………… = ……………….…………………………
= ……………….…………………………
= ……………….………………………… 1.18)  a 10 x  a 12 x  a 18 , b ≠ 0
= ……………….………………………… b b b

= ……………….…………………………

= ……………….…………………………

= ……………….…………………………

16

1.19) 23.24 1.20) 32.34.32

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.21) 52.510 1.22) 55.52.53.54

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.23) (-10).(-10)2.(-10)5 1.24) (-7)10.(-7)12.(-7)14

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.25)  5 3 x  5 5 x  5 4 1.26)  1 20 x  1 4 x  1 
7 7 7 3 3 3
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.27)  − 4 10 x  − 4 6 x  − 4 2 1.28) (0.12)3.(0.12)4.(0.12)4
5 5 5
= ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….…………………………

1.29) (0.256)8.(0.256)2 1.30) (0.2)4.(0.2)5.(0.3)2.(0.3)4
= ……………….…………………………
= ……………….………………………… = ……………….…………………………
= ……………….…………………………
= ……………….…………………………

= ……………….…………………………

2. ถ้า m และ n เปน็ จำนวนเตม็ บวก จงเขยี นจำนวนต่อไปน้ีในรูปเลขยกกำลงั ท่ีมเี ลขชีก้ ำลงั

เป็นจำนวนเต็มท่ีมากกวา่ 1

2.1) X3.X3m = ……………….……… = ……….…………

2.2) A2m.A3m = ……………….……… = ……….…………

2.3) Am.A2n = ……………….……… = ……….…………

2.4) An.A2.A2n = ……………….……… = ……….…………

17

2.5) B4m.B5m.B6m = ……………….……… = ……….…………
2.6) Xm+7.Xm+8 = ……………….……… = ……….…………
2.7) A2n.A3n+1 = ……………….……… = ……….…………
2.8) P3n+2.P2n+2 = ……………….……… = ……….…………
2.9) (AB)2n(AB)2 = ……………….……… = ……….…………
2.10) (AB)m(AB)m = ……………….……… = ……….…………
2.11) (XY)2n+8(XY)10n+2 = ……………….……… = ……….…………
2.12) (MN)(MN)4t(MN)2t = ……………….……… = ……….…………

3. จงทำใหอ้ ยู่ในรูปอย่างง่าย

3.1) (A2B3)(A4B7) 3.2) (M6N)(M2N8)
= ……………….…………………………
= ……………….………………………… = ……………….…………………………
= ……………….…………………………
= ……………….…………………………
3.3) (25M4)(23M2)
= ……………….………………………… = ……………….…………………………
= ……………….…………………………
= ……………….………………………… 3.4) (83x2y4)(8x7y6)

3.5) (3x2)(5y)(2xy) = ……………….…………………………
= ……………….…………………………
= ……………….………………………… = ……………….…………………………
= ……………….…………………………
= ……………….…………………………
3.7) (3a)(2a2b)(4b2)
= ……………….………………………… 3.6) (4xy)(32x2y4)
= ……………….…………………………
= ……………….………………………… = ……………….…………………………

3.9) (3x)(7x5)(-x3) = ……………….…………………………
= ……………….…………………………
= ……………….………………………… = ……………….…………………………
= ……………….…………………………
= ……………….………………………… 3.8) (-x4y6z3)(-7x5yz2)
= ……………….…………………………
= ……………….…………………………

= ……………….…………………………

= ……………….…………………………

3.10) (-3x)2  1 x4  (9x7)
6

= ……………….…………………………

= ……………….…………………………

= ……………….…………………………

= ……………….…………………………

18

3.11) (10y14)  − 1 y 92  (-y9) 3.12) (-12x18)(-6x32)(-2x)
5
= ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………
= ……………….…………………………
= ……………….…………………………

3.13) (-4a3)(3a5)  − 1  3.14) (-64x)  1 x 3   − 1 x 7 
4 16 2
= ……………….…………………………
= ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………
= ……………….…………………………
= ……………….…………………………

3.15) (-p5)  − 1 p  (-8p6)  − 1 p9  3.16) (-0.04x5)(0.10x6)(-1.000x7)
4 3
= ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

3.17)  − 1 a4  (-3a5)(-2a7)(10a) 3.18)  − 1 b20  (10b23)  1 b57 
15 5 2

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………
= ……………….…………………………
3.19) (82x + y)(8x + 3y) = ……………….…………………………

3.20) (-2)2m + 10 (-2)m + n - 10

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………
= ……………….…………………………
3.21) (512.53)(52) = ……………….…………………………

3.22) (512)(53.52)

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

19

3.23) (  x + 2 )(  2 – x ) 3.24) ( ep + q )( ep – q )
= ……………….………………………… = ……………….…………………………
= ……………….………………………… = ……………….…………………………
= ……………….………………………… = ……………….…………………………
= ……………….………………………… = ……………….…………………………

******************************************

3.3.2 การหารเลขยกกำลังทมี่ ีฐานเดยี วกนั และเลขชกี้ ำลังเป็นจำนวนเตม็ บวก

บทนยิ าม : ถ้า a แทนจำนวนใด ๆ m และ n เปน็ จำนวนเต็มบวก

am  an = am – n เม่ือ m > n

am  an = 1 เม่อื m < n
an−m เม่อื m = n
am  an = 1

ตัวอยา่ งที่ 1 จงหาคา่ ของ 35  33

วธิ ที ำ 35  33 = 3x3x3x3x3 หรอื = 35 - 3
3x3x3 = 32

= 3x3

= 32

ตอบ 32  34 = 32

ตวั อยา่ งที่ 2 จงหาค่าของ 42  44

วิธีทำ 42  44 = 4 x4 หรอื = 1
4x4x4x4 44−2
=1 1
4x4 = 42

= 1
42
1
ตอบ 42  44 = 42

20

ตัวอยา่ งท่ี 3 จงหาค่าของ 54  54

วธิ ีทำ 54  54 = 5x5x5x5 หรือ = 54
5x5x5x5 54
1 1
= 1 = 1

=1 =1

ตอบ 54  54 = 1

แบบฝกึ ทักษะที่ 4

การหารเลขยกกำลงั ที่มีฐานเดียวกนั และเลขช้กี ำลังเปน็ จำนวนเต็มบวก

1. จงเขียนให้อยู่ในรูปอย่างงา่ ย

1.1) A9  A7 1.2) A8  A2

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.3) 810  84 1.4) (-2)12  (-2)5

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.5) (0.5)5  (0.5) 1.6) (12.5)12  (12.5)11

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.7)  22 6  3 1 4 1.8) (1.5)9   3 6
7 7 2
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

21

1.9)  − 5 40 ( −2.5)10 1.10) 5  3
2
= ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.11) ( c15  c6 )  c3 1.12) c15  ( c6  c3 )

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.13) 72 x74 x78 1.14) (−5)5 x (−5)10 x (−5)4
73 x75 (−5)4 x (−5)7
= ……………….…………………………
= ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.15) (−0.2)12 x(−0.2)9 1.16) (0.5)9 x(0.5)6 x(0.5)7
(−0.2)8 x(−0.2)6 (0.5)8 x(0.5)10
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.17) (−0.5)25 1.18) (−0.7)11 x(−0.7)19
(−0.5)8 x(−0.5)6 x(−0.5)4 (−0.7)6 x(−0.7)10 x(−0.7)5
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.19) (−0.9)3 x(−0.9)9 x(−0.9)14 1.20) (−1.2)15 x(−1.2)13
(−0.9)2 x(−0.9)5 x(−0.9)8 (−1.2)8 x(−1.2)6 x(−1.2)10
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

22

( )1.21)19 ( ) ( )1.22)35 3 6
4 4
2 x
13
( )2 = ………………………… 3( ) ( )4 4433 = …………………

x

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.23) 36m x34m 1.24) 28m x212m
35m 23m x26m
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.25) 712n x79n x76n 1.26) 10n x103n x1011n x107n
74n x78n 102n x104n x1010n
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.27) 64 x8 1.28) 32a4b3
16 x 4 4a2b
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

23

1.29) 256a8b6 1.30) a10 b8 c 6
128a4b3 a5b4c3
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.31) (a2b2c2 )x(a5b4c3 ) 1.32) (a8b10c6 ) x(a4b5c3 )
(a4b3c2 ) (a2b4c2 )x(a5b6c5 )
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.33) (a12b10c8 ) x (a10b5c3 ) 1.34) a3n+2
(a6b6c6 ) x(ab4c5 ) a 2 n+1
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

1.35) (ab)2n+5 (ab)3n+6 1.36) (ab)5n+4 (ab)5n+5
(ab)n+4 (ab)3n+3
= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

= ……………….………………………… = ……………….…………………………

********************************************

24

3.4 สมบตั ิของเลขยกกำลัง

3.4.1 เลขยกกำลงั ท่ีมเี ลขช้ีกำลังเปน็ ศนู ย์

บทนยิ าม : ถา้ a แทนจำนวนใด ๆ ทไี่ ม่ใชศ่ ูนย์ แล้ว a0 = 1

ตวั อยา่ งที่ จงหาผลลัพธข์ องเลขยกกำลงั ต่อไปน้ี

1.) 30 =1
วธิ ีทำ 30
ตอบ 30 = 1

2.) ( a + b )0
วิธีทำ ( a + b )0 = 1
ตอบ ( a + b )0 = 1

3.) 23 x 20

วธิ ีทำ 23 x 20 = 23 x 1

= 23

ตอบ 23 x 20 = 23

4.) a3 xa4
a7
a3 xa4 a7
วิธีทำ a7 = a7

= a7 – 7

= a0

=1

ตอบ a3 xa4 =1
a7

25

แบบฝึกทกั ษะท่ี 5

เลขยกกำลงั ที่มเี ลขช้ีกำลงั เป็นศูนย์

2. จงหาผลลัพธ์ของเลขยกกำลงั ตอ่ ไปนี้

1.1) 20 = ………………..……. 1.2) 30 = ………………..…….
= ………………..…….
1.3) 40 = ………………..……. 1.4) 50 = ………………..…….
= ………………..…….
1.5) 60 = ………………..……. 1.6) 70 = ………………..…….
= ………………..…….
1.7) 80 = ………………..……. 1.8) 90 = ………………..…….
= ………………..…….
1.9) 100 = ………………..……. 1.10) 200 = ………………..…….
= ………………..…….
1.11) (-2)0 = ………………..……. 1.12) (-3)0 = ………………..…….
= ………………..…….
1.13) (-4)0 = ………………..……. 1.14) (-5)0 = ………………..…….
= ………………..…….
1.15) (-6)0 = ………………..……. 1.16) (-7)0 = ………………..…….
= ………………..…….
1.17) (-8)0 = ………………..……. 1.18) (-9)0
= ………………..…….
1.19) (-10)0 = ………………..……. 1.20) (-20)0
= ………………..…….
1.21) (0.1)0 = ………………..……. 1.22) (0.2)0
= ………………..…….
1.23) (0.3)0 = ………………..……. 1.24) (0.4)0
= ………………..…….
1.25) (0.5)0 = ………………..……. 1.26) (0.6)0

1.27) (0.7)0 = ………………..……. 1.28) (0.8)0

1.29) (0.9)0 = ………………..……. 1.30) (1.2)0

1.31)  1 0 = ………………..……. 1.32)  1 0
2 = ………………..……. 3
1.33) 0 = ………………..……. 1.34) 0
 2 = ………………..…….  3
3 0 = ………………..……. 4
1.35) 1.36) 0
 2  − 1
5 2

1.37)  − 3 0 1.38)  − 4 0
4 5

1.39)  − 2 0 1.40)  − 3 0
7 5

26

3. จงหาผลลพั ธข์ องเลขยกกำลังตอ่ ไปน้ี เมอ่ื ฐานไมเ่ ท่ากับศนู ย์

2.1) 23 x 20 2.2) 33 x 30

= ................................................................ = ................................................................

= ................................................................ = ................................................................

= ................................................................ = ................................................................

2.3) 50 x 54 2.4) 70 x 75

= ................................................................ = ................................................................

= ................................................................ = ................................................................

= ................................................................ = ................................................................

2.5) 23 x 24 x 20 2.6) 32 x 34 x 30

= ................................................................ = ................................................................

= ................................................................ = ................................................................

= ................................................................ = ................................................................

2.7) 34 x 35 x 30 2.8) 52 x 54 x 50

= ................................................................ = ................................................................

= ................................................................ = ................................................................

= ................................................................ = ................................................................

2.9) A2 x A0 2.10) A4 x A0

= ................................................................ = ................................................................

= ................................................................ = ................................................................

= ................................................................ = ................................................................

2.11) B3 x B4 x B0 2.12) X6 x X8 x X0

= ................................................................ = ................................................................

= ................................................................ = ................................................................

= ................................................................ = ................................................................

2.13) (ab)3 x (ab)0 2.14) (ab)0 x (ab)5

= ................................................................ = ................................................................

= ................................................................ = ................................................................

2.15) (a + b )2(a + b)0 2.16) (a + b)(a + b)0

= ................................................................ = ................................................................

= ................................................................ = ................................................................

= ................................................................ = ................................................................

27

2.17) 24 = ........................................... 2.18) 26 = ...........................................
24 26 = ...........................................
= ...........................................

= ........................................... = ...........................................

2.19) 37 = ........................................... 2.20) 58 = ...........................................
37 = ........................................... 58 = ...........................................

= ........................................... = ...........................................

2.21) 710 = ........................................... 2.22) 1012 = ...........................................
710 1012
= ........................................... = ...........................................

= ........................................... = ...........................................

2.23) a5 = ........................................... 2.24) a9 = ...........................................
a5 a9 = ...........................................
= ...........................................

= ........................................... = ...........................................

2.25) b8 xb7 = ........................................... 2.26) b4 xb6 = ...........................................
b4 xb11 b10
= ........................................... = ...........................................

= ........................................... = ...........................................

= ........................................... = ...........................................

2.27) m10 m20 = ........................................... 2.28) p2 p3 p5 = ...........................................
m18 m12 p2 p8
= ...........................................
= ...........................................

= ........................................... = ...........................................

= ........................................... = ...........................................

2.29) y 5 y12 y13 = ......................................... 2.30) A2n + 3A-2n - 3= ......................................
y10 y 20 = .......................................

= ........................................... = .......................................

= ........................................... = .......................................

= ........................................... = .......................................

28

2.31) B4m + 2 x B-3m + 3x B-m - 5 2.32) b2n+3 = ...........................................
= ..................................................... b2n+3
= ....................................................... = ...........................................

= ....................................................... = ...........................................

= ....................................................... = ...........................................

2.33) x 4n−3 = ........................................... 2.34) x x2n+4 3n−1 = .....................................
x 4n−3 x 5n+3
= ........................................... = ......................................

= ........................................... = ......................................

= ........................................... = ......................................

= ........................................... = ......................................

2.35) x x2n+4 3n+6 = .................................... 2.36) 2 x n+4 x 2n−3 = ...................................
x 5n+10 x 3n+1
= ........................................... = ...........................................

= ........................................... = ...........................................

= ........................................... = ...........................................

= ........................................... = ...........................................

= ........................................... = ...........................................

3 x 3n+5 2 x 2 n−1 4 x 4n+9 3 x n−3
x 5n+4 x 5n+6
2.37) = ................................ 2.38) = .................................

= ........................................... = ...........................................

= ........................................... = ...........................................

= ........................................... = ...........................................

= ........................................... = ...........................................

= ........................................... = ...........................................

29

3.4.2 เลขยกกำลังทีม่ เี ลขช้กี ำลังเป็นลบ

บทนยิ าม : ถ้า a แทนจำนวนใด ๆ ทไ่ี ม่ใชศ่ นู ย์ และ n เป็นจำนวนเต็มบวก

แลว้ a-n = 1 และ 1 = an
an a−n

ตัวอย่างที่ จงหาผลลัพธ์ของเลขยกกำลังต่อไปน้ี

1.) 3-1

วิธีทำ 3-1 = 1
31
1
ตอบ 3-1 = 3
2.) 2-2

วิธที ำ 2-2 = 1
22
1
= 4

ตอบ 2-2 = 1 หรือ 1
22 4
37 x39
3.) 38 x320

วธิ ีทำ 37 x39 = 37+9
38 x320 38+20
316
= 328

= 316 – 28

= 3-12

= 1
312
37 x39
ตอบ 38 x320 = 1
312

4.) a2 xa3 30
a6 = ( -3 )4
a2 xa3 a2+3
วิธที ำ a6 = a6

= a5
a6
= a5 - 6

= a-1

= 1
a1
a2 xa3
ตอบ a6 = 1
a1
1
5.) 2−2

วิธที ำ 1 = 22
2−2
1
ตอบ 2−2 = 22

6.) 1
(−3)3 x(−3)−7
1 1
วธิ ีทำ (−3)3 x(−3)−7 = ( −3) 3+( −7 )

= 1
(−3)−4
1
ตอบ (−3)3 x(−3)−7 = ( -3 )4

7.) 1
( 1 )−2
2
1 = (1)2
วิธที ำ 2
1 −2
( 2 )

ตอบ 1 = ( 1 )2
2
( 1 ) −2
2

31

8.) 1
a−4m xam
1 1
วิธีทำ a−4m xam = a ( −4 m )+ m

= 1
a −3 m
= a3m

ตอบ 1 = a3m
a−4m xam

แบบฝกึ ทกั ษะที่ 6

เลขยกกำลงั ท่ีมีเลขช้กี ำลงั เป็นลบ

1. จงหาใหอ้ ยใู่ นรูปอยา่ งง่ายและเลขช้กี ำลงั เป็นบวก

1.1) 5-6 1.2) 7-10

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.3) 10-8 1.4) 3-4

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.5) 23 x 2-5 1.6) 3-10 x 37

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.7) 56 x 54 x 5-14 1.8) 5-18 x 57 x 56

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

32

1.9) 75 1.10) 1018

79 1025
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.11) 310 x311 x312 1.12) (−2)4 (−2)3

315 x316 x317 (−2)10
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.13) (−5)0 (−5)9 (−5)6 1.14) (−3)9 (−3)6 (−3)7

(−5)10 (−5)12 (−3)13 (−3)17
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.15) (0.2)5 (0.2)5 1.16) (0.1)6 (0.1)5

(0.2)6 (0.2)8 (0.1)18
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

33

1.17) ( 3 )3 x( 3 )5 x( 3 )6 1.18) ( − 1 )3 x ( − 1 )2 x (− 1 )6
444 5 5 5
3 3 3 1 1 1
( 4 )4 x ( 4 )7 x ( 4 ) 8 ( − 5 )8 x (− 5 )4 x (− 5 )0

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.19) (− 3 ) 6 x ( − 3 )6 x ( − 3 ) 4 1.20) ( − 4 ) 2 x ( − 1 ) 2 x ( − 1 ) 6
5 5 5 5 5 5
3 3 3 4 4 4
(− 5 ) 2 x ( − 5 )10 x ( − 5 )8 (− 5 )5 x ( − 5 ) 8 x ( − 5 ) 6

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2. จงทำให้อยู่ในรปู อยา่ งงา่ ยและใหเ้ ลขชก้ี ำลงั เป็นบวก โดยที่ฐานของเลขยกกำลังไมเ่ ท่ากับ
ศูนย์ เม่ือ m , n เปน็ จำนวนเตม็

2.1) b-1 2.2) b-15

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.3) a5 2.4) a9

a6 a12
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

34

2.5) a15 2.6) a4 xa6

a20 a5 xa7
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.7) bn 2.8) d2n

b2n d5n
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.9) x m x2m 2.10) x 3m x 4m

x5m x 5m x 3m
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.11) x n+1 2.12) y m+n+1

x 3n+1 y 2m+n+1
= ............................................................ = ............................................................
= ............................................................ = ............................................................
= ............................................................ = ............................................................
= ............................................................ = ............................................................

2.13) a3m−2n+1 2.14) a2ma3m+2

a5m−2n+1 a7m+2
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

35

2.15) (ab)3m+2 2.16) (ab)3m+4 (ab)2m+1

(ab)4m+2 (ab)7m+5
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.17) (a + b)m+1 2.18) (a + b)0 (a + b)5m+2

(a + b)3m+1 (a + b)8m+2
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.19) (a + b)2 (a + b)m+n 2.20) (a + b)2n−2 (a + b)n+3

(a + b)1+n (a + b)1+3m (a + b)n+4 (a + b)4n−3
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

3. จงหาให้อยใู่ นรปู อยา่ งง่ายและเลขชกี้ ำลังเปน็ บวก

3.1) 1 3.2) 1

2−7 3−15
= ............................................................ = ............................................................

= ............................................................ = ............................................................

36

3.3) 1 3.4) 1

74 x7−6 2−8 x23
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

3.5) 1 3.6) 1

38 x36 x3−15 52 x5−10 x56
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

3.7) 1 3.8) 1

(−2)−12 (−3)−7
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

3.9) 1 3.10) 1

(−3)6 x(−3)−10 (−5)−12 x(−5)5 x(−5)4
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

3.11) 1 3.12) 1

(−7)0 x(−7)−18 x(−7)10 (−3)7 x(−3)−13 x(−3)4
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

37

3.13) (0.1)0 3.14) (0.2)0

(0.1)3 x(0.1)−4 (0.2)6 x(0.2)−10
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

3.15) ( 3 )3 x( 3 )−3 3.16) ( 3 )0
44 4
( 3 )−5
4 ( 3 )3 x( 3 )7 x( 3 )−15
444
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

( ) ( )3.17) − 1 4 − 1 −4 ( ) ( )3.18) − 1 4 − 1 −4
22 22
5 8 −14 5 8 −14
1( ) ( ) ( )− 1 1 ( ) ( ) ( )−1 1 1
2 − 2 − 2 2 − 2 − 2

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

38

3.19)  4 3  4   4 −4 ( ) ( ) ( )3.20)323 3 3 −5
5 5 5 7 7 7
− − − − − −

 4 0  4 5  4 −8 ( ) ( ) ( )−38 3 6 3 −17
5 5 5 7 7 7
− − − − −

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

4. จงทำให้อยู่ในรูปอย่างงา่ ยและให้เลขชกี้ ำลงั เป็นบวก โดยท่ีฐานของเลขยกกำลังไม่เท่ากับ
ศนู ย์ เม่อื m , n เปน็ จำนวนเต็ม

4.1) 1 4.2) 1

a−1 a−2
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

4.3) 1 4.4) 1

a−4 a−8
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

4.5) 1 4.6) 1

a −2 m a −3 m
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

39

4.7) 1 4.8) 1

a−4m xam bn xb−3n
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

4.9) 1 4.10) 1

b−6n xb−7n b5m xb−6m
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

4.11) d0 4.12) d0

d2m xd−7m d−9m xd4m
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

4.13) X n X −n 4.14) X 2n X −2n

X 7m X −10m X 3n X −4n
= ............................................................ = ............................................................
= ............................................................ = ............................................................
= ............................................................ = ............................................................
= ............................................................ = ............................................................

4.15) y n−3 y −n+3 4.16) y −n+1 y n−1

y 3 y −3n−3 y y3m−4 −6m+4

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

40

4.17) x 2n x −n x −n 4.18) x1−n x n−1

x x3m−1 −5m+1 x 4m x −6m−2 x 2
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

4.19) (ab)m (ab)−m 4.20) (ab)0

(ab)−3m (ab)4m (ab)−7m
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

3.4.3 สมบตั ิของเลขยกกำลงั

บทนยิ าม : ถ้า a , m , n แทนจำนวนใด ๆ แล้ว ( am )n = am x n

ตวั อย่างท่ี จงหาผลลพั ธ์ของเลขยกกำลังต่อไปน้ี

1.) ( 2 2)2 = 22 x 2
วิธีทำ ( 2 2)2 = 24

ตอบ ( 2 2)2 = 24

2.) ((0.1)3)3

วิธีทำ ((0.1)3)3 = (0.1)3 x 3

= (0.1)9

ตอบ (0.1)3)3 = (0.1)9

3.) (( 1 )2)4 41
2
1 1 = ………………..…….
วธิ ที ำ (( 2 )2)4 = ( 2 )2 x 4 = ………………..…….
= ………………..…….
= ( 1 )8 = ………………..…….
2 = ………………..…….
1 1 = ………………..…….
ตอบ (( 2 )2)4 = ( 2 )8 = ………………..…….
= ………………..…….
4.) ((( − 3 )2)-2)-2
5
3 3
วิธที ำ ((( − 5 )2)-2)-2 = (( − 5 )2)-2 x -2

= (( − 3 )2)4
5
3
= ( − 5 )2 x 4

= ( − 3 )8
5
3 3
ตอบ ((( − 5 )2)-2)-2 = ( − 5 )8

แบบฝกึ ทักษะที่ 7

สมบัตขิ องเลขยกกำลัง

1.) จงหาผลลพั ธข์ องเลขยกกำลังต่อไปนี้

1.1) ( 24 )3 = ………………..……. 1.2) ( 3 3 )4
1.3) ( 54 )5 = ………………..……. 1.4) ( (-2)3 )2
1.5) ( (-5)6 )5 = ………………..……. 1.6) ( (-7) 3 )5
1.7) ((0.1)2)3 = ………………..……. 1.8) ((0.2)5)3
= ………………..…….
= ………………..…….
= ………………..…….
= ………………..…….

42

1.9) ((0.3)5)4 = ………………..……. 1.10) ((0.5)6)3 = ………………..…….
1.11) (( 1 )3)4 = ………………..……. 1.12) (( 2 )-3)-3 = ………………..…….
= ………………..……. = ………………..…….
2 = ………………..……. 3 = ………………..…….
1.13) (( 3 )2)3 = ………………..……. 1.14) (( − 2 )0)3 = ………………..…….
= ………………..……. = ………………..…….
4 = ………………..……. 3 = ………………..…….
1.15) ((( − 4 )-1)-2)3 = ………………..……. 1.16) ((( − 2 )-1)-1)2 = ………………..…….
= ………………..……. = ………………..…….
5 = ………………..……. 7 = ………………..…….
= ………………..……. = ………………..…….

2.) จงทำเลขยกกำลังต่อไปน้ีให้อยู่วนรปู อย่างงา่ ย และให้เลขชี้กำลังเปน็ จำนวนเต็มบวก
โดยที่ฐานของเลขยกกำลังต้องไมเ่ ท่ากับศูนย์ เมือ่ m , n เปน็ จำนวนเต็ม

2.1) ( a 2 ) 3 2.2) ( a3 ) 3

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.3) ( a3 )4 2.4) ( a4 )2

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.5) ( a-2 )-3 2.6) ( a-3 )-5

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.7) ( b-3 )n 2.8) ( b-2 )-2n

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

43

2.9) ( b-4 )-2n 2.10) ( b3m )2

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.11) ( b-m )-1 2.12) ( b2 )4m

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.13) ( an + 1 )2 2.14) ( an + 3 )3

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.15) ( a2n + 1 )2 2.16) ( x3n – 2 )3

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.17) ( b3n – 1 )4 2.18) ( b2 )n + 1

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.19) ( a3 )n + 2 2.20) ( b5 )2n – 1

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

44

3.4.4 เลขยกกำลัง
บทนิยาม : ถ้า a , b , m , n และ k แทนจำนวนใด ๆ แลว้ ( am bn )k = amkbnk

ตัวอยา่ งที่ จงหาผลลัพธ์ของเลขยกกำลังต่อไปนี้

1.) ( 2 x 3 )2

วิธที ำ ( 2 x 3 )2 = 22 x 32

ตอบ ( 2 x 3 )2 = 22 x 32

2.) [(-2) x (-3)]3

วธิ ีทำ [(-2) x (-3)]3 = (-2)3 x (-3)3

ตอบ [(-2) x (-3)]3 = (-2)3 x (-3)3

3.) [(0.1)2 x (0.2)3]2

วิธีทำ [(0.1)2 x (0.2)3]2 = (0.1)2 x 2 x (0.2)3 x 2

= (0.1)4 x (0.2)6

ตอบ [(0.1)2 x (0.2)3]2 = (0.1)4 x (0.2)6

4.) [( 1 )3 x( 1 )2]3
2 3
1 1 1 1
วิธที ำ [( 2 )3 x( 3 )2]3 = ( 2 )3 x 3 x( 3 )2 x 3

= ( 1 )9 x( 1 )6
2 3
1 1 1 1
ตอบ [( 2 )3 x( 3 )2]3 = ( 2 )9 x( 3 )6

45

แบบฝกึ ทกั ษะท่ี 8

สมบตั ิของเลขยกกำลัง ( 2 )

1. จงหาใหอ้ ย่ใู นรปู อยา่ งง่ายและเลขชีก้ ำลงั เปน็ บวก

1.1) ( 2 x 3 )3 1.2) ( 22 x 3 )2

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.3) ( 23 x 52 )3 1.4) ( 54 x 73 )4

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.5) [(-2)4 x (-3)4]2 1.6) [(-2)-5 x (-3)-2]-4

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.7) [(-5)-3 x (-7)-2]-2 1.8) [(0.3)-2 x (0.5)-3]-2

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.9) [(-0.1)3 x (-0.2)3]3 1.10) [(-0.3)-4 x (-0.5)-2]-3

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.11) [( 1 )-5 x ( 2 )-3]-3 1.12) [(- 1 )2 x (- 1 )3]4
27 23
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.13) [(- 1 )0 x (- 2 )2 x (- 4 )3]3 1.14) [(- 3 )-3 x (- 3 )3 x (- 2 )-4]-2
235 4 45
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

46

1.15) ( a-2b-3 )-2 1.16) ( 22 x 33 x a x b4 )3

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.17) ( a3b4 )m 1.18) ( ab2 )m

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.19) ( a-6b-1 )-2m 1.20) ( a-3b-2 )- m

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.21) ( a2b )n 1.22) ( x2y2 )n

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

3.4.5 เลขยกกำลัง

บทนยิ าม : ถา้ a , b , n แทนจำนวนใด ๆ แลว้ ( a )n = an
bn
b

ตวั อยา่ งที่ จงหาผลลัพธ์ของเลขยกกำลงั ต่อไปน้ี

1.) ( 2 )2
3
22
วิธีทำ ( 2 )2 = 32
3

ตอบ ( 2 )2 = 22
3 32

47

2.) ( a )2n
b
a2n
วิธที ำ ( a )2n = b2n
b a2n
b2n
ตอบ ( a )2n =
b

แบบฝกึ ทกั ษะที่ 9

สมบัตขิ องเลขยกกำลัง ( 3 )

1. จงหาให้อยูใ่ นรปู อย่างง่ายและเลขช้กี ำลังเปน็ บวก

1.1) ( 1 )3 1.2) ( 6 )-4
3 11
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.3) ( 3 )-6 1.4) ( 2 )-5
7 7
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.5) ( - 2 )2 1.6) ( - 1 )3
3 3
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.7) (- 7 )6 1.8) (- 9 )7
8 10
= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

48

1.9) −  4 4 1.10) −  6 5
5 11
= ............................................................
= ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.11) −  7 3 1.12) − − 4 6
9 5

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.13)  a 4 1.14)  a −2
b b

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.15)  a 3 n 1.16)  a 4 n
b b

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.17)  a −2n 1.18)  x 4n
b  y 
= ............................................................
= ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

1.19) − x −3n 1.20) − x −4 n
y  y 

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

49

2. จงหาใหอ้ ยใู่ นรูปอย่างง่ายและเลขชกี้ ำลังเปน็ บวก

2.1)  2−2 x33 2 2.2)  2−3 x3−1 3
 54   72 

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.3)  3−2 x52 −2 2.4)  2−1 x30 x5−2 −1
 23 x7−1   3−3 x74 

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.5)   2−1 x34 x5−2 −1 −2 2.6)    2−3 x3−2 −1 −1 −1
  3−2 x73      5−1   
= ............................................................  

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................

2.7)  a4 −1 2.8)  a2 xb 2
 b2   a4 

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

2.9)  a3 xb2 −2 2.10)  a−1b−3 −1
 a−4 xb3   a−4 xb−5 

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................

= ............................................................ = ............................................................