Grade 8 Book 1 Unit 1 and 2

SARASAS AFFILIATED SCHOOLS
Mathematics Grade 8

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Foreword:
By Dr. Chamrat Nongmak

When talking about bilingual education in private elementary and secondary schools in Thailand, discussion
invariably tends to focus on Sarasas Affiliated Schools, a large group of private schools that was formerly owned
and administered by the late Mr. Peboon Yongkamol. Mr. Peboon, a well-known and respected educator, was a
leading pioneer of the bilingual education concept here in Thailand. These schools provide a benchmark and a
showcase for bilingual education in Thailand. Many school proprietors, directors, and managers make it a point to
visit Sarasas Affiliated Schools seeking advice and guidance from a successful leader in the field of bilingual
education.

Sarasas programs for teaching students both in English and in Thai have been carefully planned and
developed step by step. One of the toughest challenges Sarasas has had to face has been finding the necessary
texts and materials for teaching Thai subjects in English. This is because a bilingual school, like any other Thai school,
needs government approval to operate and must teach the curriculum mandated by the Ministry of Education.
Unfortunately, the materials needed for teaching Thai subjects in English are not readily available. Publishers do
little to support this need because the concept is still fairly new and the demand is small. This has forced Sarasas to
create its own texts and materials.

Sarasas has been more fortunate though, than most other schools struggling to develop their own bilingual
program. It already has bilingual education experience, a qualified staff and the resources necessary to produce its
own high-quality textbooks and materials. The staff, of course, is the most important ingredient. Sarasas’ large,
well-seasoned staff of highly competent, well qualified and dedicated teaching professionals, both Thai and native
speakers of English, gives it a unique advantage over others attempting to produce these much-needed materials.

Even so, it has not been an easy task. The materials have been produced through the careful process of
research and development. Revision or total change is done when those materials are proven to be inadequate.
The people involved in developing these textbooks or workbooks are classroom teachers. They are the ones who
will have to teach with the same materials they have helped to create. Knowing this gives them a greater incentive
to do it right.

The teams of classroom teachers responsible for classroom material development operate under the close
supervision of the Sarasas Board of Executives.

One of the roles of the board is to assure proper quality control by collecting and carefully evaluating the
actual work done by students using the new materials. They make certain that the students are able to understand
the material and use them effectively. One of the things they check for, is whether the level of English used in the
books is appropriate for the students’ level of English. The board also makes certain that the content of the material
is in accordance with the approved syllabus. The Mathematics book has been developed through this process.

This book is not only useful for teachers but also for parents and guardians. Its purpose is to encourage and
support students in their quest for knowledge and understanding of Mathematics, as well as stimulate their interest
and improve their skills. Each unit provides an introduction to the material, a list of new words the student will
encounter, problem solving examples to help the student better understand how to do the work, oral exercises,
and a review of what was learned.

This new Maths material will provide students with a greater international perspective and understanding
of Mathematics and Mathematics terminology which will benefit them greatly in later life. This said, I would like to
take this opportunity to thank those responsible for developing this material and offer my sincere congratulations
for a job well done.

Chamrat Nongmak, PhD.

Former Deputy Permanent Secretary
The Ministry of Education

PREFACE

This Mathematics textbook has been developed for bilingual learners
based on the B.E. 2560 (A.D. 2017) revised version of Thailand’s Basic Education
Core Curriculum B.E. 2551 (A.D. 2008). The content of this book follows the Basic
Mathematics for Grade 8 Book written by The Institute of Academic
Development.

This textbook aims to enhance students’ abilities needed in the 21st
century. These include analytical skills, problem solving skills, creativity and
collaboration skills.

At the beginning of each unit of this textbook, a useful vocabulary list is
provided to learners. Examples demonstrate to students how to apply the
information they have learnt to solve related problems. Exercises and activities
follow that allow for immediate practice. At the end of each unit, a summary of
the key learning concepts along with a revision exercise is provided to help
students consolidate what they have learnt.

Author,
Sarasas Affiliated Schools

March, 2022









UNIT 1

Powers and Indices

BITS FROM THE PAST…

Euclid(u-clid), the person on the right, discovered
the concept of exponents, which he used in his geometric
equations. He used the term “power” to represent what
we know today, how many times a number is multiplying

by itself.
Later, Archimedes, picture on the left, generalized the idea of powers.
\ He also discovered the first Scientific Notation.
The mathematicians from the Islamic golden age had discovered the
uses of powers of two and three while working in algebra.
In 1484, Nicholas Chuquet, a French mathematician, was the first
person to use exponential notation. He also acknowledged zero and negative
numbers as exponents.

Mathematical Terms indices (เลขชก้ี ำลงั )
integer (จำนวนเตม็ )
accumulate (รวบรวม) interest (ดอกเบย้ี )
applicable (นำไปประยุกตใ์ ชไ้ ด)้ law (กฎหมำย)
base (ฐำน) numerator (เลขเศษ)
correspond (สอดคลอ้ งกนั ) per annum (ต่อปี)
denominator (ตวั เลขทอ่ี ย่สู ว่ นล่ำงของเศษส่วน) principal (เงนิ ตน้ )
denote (หมำยถงึ ) product (ผลคณู )
evaluate (ประเมนิ ) rate of interest (อตั รำดอกเบย้ี )
except (ยกเวน้ ) real number (จำนวนจรงิ )
exponent (เลขชก้ี ำลงั ) reciprocal (ซง่ึ กนั และกนั )
exponential form (เลขยกกำลงั )
index form (เลขยกกำลงั ) Mathematics Book 1

1

1. Introduction to Indices

What are Indices?

Indices are used as the shorthand way of writing multiplications of the same

item several times. “Powers”, “Indices” and “Exponents” are all descriptions of the

exact same thing. index, power or exponent

base

The base is the number that will be multiplied to itself. It can be any real
number. The index (singular form of indices) is the number of times you multiply
the base to itself.

There are two ways to read the numbers above:
“seven to the fourth power” or “seven to the fourth”

NOTE: NEVER MULTIPLY THE BASE TIMES THE EXPONENT. (74 ≠ 7 4)

More examples: Expanded Form Base Power
Index/Exponential 5×5×5 5 3
−2 4
Form (−2) × (−2) × (−2) × (−2)
53

(−2)4

Therefore, we can conclude that:
For any real number and for any positive integer ,
= × × × … × ( to times)

Exercise 1A

1) Express each product as a power.

a) 2 × 2 × 2 × 2 × 2

b) (−3) × (−3) × (−3) × (−3) × (−3) × (−3)

c) 2.5 × 2.5 × 2.5

d) 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2
3 3 3 3 3 3 3 3

e) ( + )( + )( + )( + )

2 Mathematics Book 1

2) Write the following numbers in index form. The base MUST be a prime.

a) 16 d) 0.0081

b) −125 e) −0.343

8 625

c) 27 f) 1 331

2. Laws of Indices

Indices are a convenient tool in mathematics to compactly denote the
process of taking a power of a number. Taking a power is simply a case of repeated
multiplication of a number with itself. Therefore, it is important to understand the
concepts as well as the laws of indices.

2.1 Multiplying Powers With Same Bases

If is a real number and and are integers, then

= + .

Note that addition of powers is applicable only when the bases are the same.

Multiply the following expressions:

a) 25 23 b) (−6)2 (−6)8

a) 25 23 = 25+3 b) (−6)2 (−6)8 = (−6)2+8
= = (− )

2.2 Dividing Powers With Same Bases

If is a real number not equal to zero and and are integers, then

− .

=

Note that subtraction of powers is applicable only when the bases are the same.

Mathematics Book 1 3

Divide the following expressions: 256
a) 715 ÷ 79
b) 215

a) 715 ÷ 79 = 715−9 256 = 256−15
= =
b) 215

2.3 Raising a Power to a Power

If is a real number and and are integers, then

( ) = ×

Note that to raise a power to another, multiply the exponent.

Example 3

Simplify the following powers: b) (−92)6
a) (53)4
b) (−92)6 = (−9)2 × 6
Solution = (− )

a) (53)4 = 53 × 4
=

2.4 The Zero Exponent

Recall that ÷ = − , for ≠ 0.
Therefore, if = , then − = − = 0.
But dividing the numerator by the denominator, that is ÷ = 1.
Therefore,

= .

Any number or algebraic expression (except zero), having a zero exponent
has a numerical value of 1.

4 Mathematics Book 1

Evaluate each expression. b) (6 + )0 + 50
a) 230

a) 230 = b) (6 + )0 + 50 = 1 + 1
=

Exercise 1B

1) Multiply the following indices.

a) 8 × 84 f) (−3)3 × (−3)2 × (−3)9
_______________________
_______________________ _______________________

_______________________

b) 73 × 76 g) 7 • 8 • 13

_______________________ _______________________
_______________________
_______________________

c) 116 × 117 h) 2.76 × 2.73 × 278
_______________________
_______________________ _______________________

_______________________

d) (0.2)4(0.2)8 i) ( )33 • ( )45 • ( )67

_______________________ _______________________
_______________________
_______________________

e) ቀ1ቁ9 • ቀ1ቁ7 • ቀ1ቁ7 j) ቀ− 11ቁ19 ቀ− 11ቁ76 ቀ− 11ቁ89

22 2 333

_______________________ _______________________
_______________________ _______________________

2) Simplify the following exponential expressions.

a) 50 e) 00
_______________________ _______________________

b) ቀ4ቁ0 f) 0
_______________________
7
_______________________

c) (−0.25)0 g) 4 0

_______________________ _______________________
d) (23 × 25)0 0 0
h) ቀ− 1 − ቀ1
_______________________ ቁ ቁ
22

_______________________

Mathematics Book 1 5

3) Divide the following exponential numbers.

a) 27 ÷ 25 f) (−0.13)67 ÷ (−0.13)67
_______________________ _______________________

_______________________ _______________________
g) 47 ÷ 44 ÷ 42
b) 811 ÷ 87
_______________________ _______________________

_______________________ _______________________

212 h) ቀ1ቁ54 ÷ ൤ቀ1ቁ27 ÷ ቀ1ቁ7൨

c) 29 3 33

_______________________ _______________________

_______________________ _______________________

d) 1717 i) ( − )109
178 ( − )76

_______________________ _______________________

_______________________ _______________________

e) ( )3 ÷ ( )2 j) (−1)43 ÷(−1)34
_______________________ (−1)9
_______________________ _______________________

_______________________

4) Simplify the following powers. f) ((−0.08)3)−6
_______________________
a) (52)5
_______________________ _______________________
_______________________ g) ሾ(2 − )13ሿ2

b) (0.24)9 _______________________
_______________________
_______________________ _______________________
h) (8990)899
c) (−2346)8
_______________________
_______________________
_______________________ _______________________

d) (5−5)7 i) ൤ቀ3ቁ11൨7

_______________________ 4
_______________________
_______________________
e) ( 2)45
_______________________
_______________________
_______________________ 1
13
j) 1
ቁ3൩
൥ቀ(−6)3

_______________________
_______________________

6 Mathematics Book 1

3. The Other Laws of Indices

3.1 Raising a Product to a Power

For any real numbers and , and for any whole number ,

( ) = ,

where ≠ 0 and ≠ 0.

Distribute the given powers: b) (0.7 × 5)4
a) (2 × 3)3

a) (2 × 3)3 = (2 × 3) • (2 × 3) • (2 × 3)
= (2 × 2 × 2) • (3 × 3 × 3)
= ×

b) (0.7 × 5)4 = (0.7 × 5)4
= . ×

Write 358 in the exponential bases form with a base of prime number.

358 = (7 × 5)8
= ×

3.2 Raising a Quotient to a Power

For any real numbers and , and for any whole number ,

ቀ ቁ =

,

where ≠ 0.

Mathematics Book 1 7

Example 7 b) ቀ− 13ቁ9

Simplify the following indices.

a) ቀ79ቁ2

Solution

a) ቀ7ቁ2 = 7 × 7 b) ቀ− 1ቁ9 = ቀ− 1ቁ9
9 9
9 3 3

= 7×7 = −
9×9

=


3.3 The Negative Power

We know that for any real number , 0 = 0− = − . But 0 = 1, therefore,


− = ,


where is a positive integer and a nonzero real number .

Note: Any quantity with a negative integer power is equal to the reciprocal
of that quantity with the corresponding positive exponent. Any factor of the
numerator of a fraction may be transferred to the denominator, or any factor of the
denominator can be transferred to the numerator if the sign of its power is
changed.

Simplify the following expressions in positive indices.

a) 3−1 b) (−5)−9 c) ቀ34ቁ−2

a) 3−1 = b) (−5)−9

= (− )

c) ቀ43ቁ−2 = ቀ ቁ

8 Mathematics Book 1

Exercise 1C

1) Distribute the given powers.

a) ( )3 e) ( )7
_______________________
_______________________

b) (3 × 8)12 f) ሾ(−2) × 7 × 11ሿ8

_______________________ _______________________

c) ቀ2ቁ5 g) ቀ ቁ4

3

_______________________ _______________________

d) ቀ ቁ9_______________________ h) ቂ (15 × 13) 3

ቃ
(2_3__×__1_7_)________________

2) Rewrite in the exponential bases form with bases of prime numbers.

a) 153 _______________________ e) 8

27 _______________________

_______________________ _______________________

b) 6312_______________________ f) 25
_______________________
49 _______________________

_______________________

c) 1431_9______________________ g) 121
_______________________
169_______________________

_______________________

d) 1152_______________________ h) 121
_______________________
289_______________________

_______________________

3) Simplify and write the answers using positive exponents only.

a) 10−1 d) ቀ79ቁ_−_1_____________________
_______________________

_______________________ _______________________
b) 15−51 ቀ− 3ቁ−13
e)
_______________________ __2_____________________

_______________________ _______________________

c) ( )−3 f) ቀ 25_ቁ_−__2___________________
_______________________

_______________________ _______________________

_______________________

Mathematics Book 1 9

4. Simplifying Indices

In order to simplify the numbers or algebraic expressions involving indices,
we have to use and combine one or more laws of indices.

1) Simplify and write the answers using positive exponents only.

25 ÷5−6 c) ( )−5 × ( 2 −3)6

a) 512 −5

3 5• (2+ ) d) ቀ8ቁ4 × ቀ2342 ቁ

b) 6 3 9

a) 25 ÷5−6 = 52 ÷5−6
512 512

52−(−6) ( ÷ = − )

= 512

= 58−12

= 5−4 ( − = )



=

b) 3 5• (2+ ) = 3• (5+2+ ) ( × = + )
6 3
6 3

3 (7+ )

= 6 3

3 (7+ −3) ( ÷ = − )
(cancelling method)
=

6
3 1 (4+ )

= 62

( + )

=

10 Mathematics Book 1

c) ( )−5 × ( 2 −3)6 = ( −5 −5) × ൫ (2×6) (−3×6) ൯

= ( −5 −5) × ( 12 −18 )

= ( −5 × 12) × ( −5 × −18)

= (−5+12) × (−5−18)

= 7 × −23

= 7 × 1
23



=

d) ቀ8ቁ4 × ቀ2324ቁ−5 = ቀ2323ቁ4 × ቀ3224ቁ−5

9

൫23൯4 (24)−5
ቀ ቁ =
= (32)4 × (32)−5

212 2−20 ( ) = ( × )

= 38 × 3−10

212+(−20) ( × = + )

= 38+(−10)

2−8

= 3−2

32 ( − = )

= 28



=

Mathematics Book 1 11

Exercise 1D

1) Find the values of the following exponential expressions. Write the

simplified answers in positive exponential forms.

a) (−2)10•(−2)8•(−2)5 = ________________________________
(−2)19 ________________________________

________________________________

________________________________

b) ቀ13ቁ6•ቀ31ቁ9 ________________________________
ቀ13ቁ8•ቀ31ቁ
= ________________________________
________________________________
________________________________

________________________________

________________________________

________________________________

7−9 •7−7• 343 = ________________________________
________________________________
c) 7−2•7−4

________________________________

________________________________

________________________________

________________________________

3−7×94 = ________________________________
________________________________
d) 275× 81−2× 243−5 ________________________________
________________________________
________________________________
________________________________

e) ቀ85 34 −−63ቁ2 × ቀ2 5 7 3 = ________________________________
25 ________________________________
ቁ

________________________________

________________________________

________________________________

________________________________

f) 3− +1 × 251− ÷ 3 −1 = ________________________________
3− −1 9− 53 −3 ________________________________

________________________________

________________________________

________________________________

________________________________

12 Mathematics Book 1

5. Applications of Indices

Exponents, indices and powers are used in lots of parts of modern world.
Indices are used in computer games, science, engineering, economics, finance,
programming and many other.

In this lesson we show several real life uses of indices.

In a warehouse, each sack of rice weighs 32 pounds. If there are 35
sacks, what is the total weight of the crate? Leave answer in index form.


.

Given: One sack of rice weighs 32 pounds.

There are 35 sacks of rice.

So,

Total weight of the crate = 32 × 35

= 32+5
= 37

Therefore, the total weight of the crate would be pounds.

Example 12

The mass of the Venus is about 0.815 times the mass of the Earth.
Given that the mass of the Earth is about 5.98 × 1024 kilograms, find the mass
of the Venus in kilogram.

Solution

Mass of the Venus: 0.815 (times the Earth’s mass)

Mass of the Venus: 5.98 × 1024 kilograms

So, Venus mass ≈ ൫8.15 ×10-1൯ × ൫5.98×1024൯

≈ (8.15×5.98) × ൫10-1×1024൯

≈ 48.737 × 1023

Therefore, the mass of the Venus is 4.8737 × 1024 kilograms.

Mathematics Book 1 13
If an amount of 2,000 baht is deposited into a savings account
compounded annually of 5%, find the value of investments after 2 years.

The problem above is related to compound interest. A compound
interest is the interest computed on the initial principal, which also include
the earned interest of previous time of a deposit or a loan.

= ( + )

When = accumulated amount (total amount)
= principal amount (initial amount)
= rate (percent in decimal form)
= time (in years)

From the given problem:

= total value of the investment
= 2,000 baht
= 5% (0.05)
= 2 years

Substitute all the given values,

= 2,000 (1 + 0.05)2
= 2,000 (1.05)2
= 2,000(1.1025)
= 2,205

Therefore, the value of the investment after 2 years is 2,205 baht.

14 Mathematics Book 1

Exercise 1D

1) There are 57 grains of sugar in a teaspoon. Mae has a plan to bake a cake.
She needs 53 teaspoon of sugar. How many grains of sugar are there?

____________________________________________________________
____________________________________________________________
____________________________________________________________
2) I_to__o_w_n_s__a_m__ic_r_o_s_co__p_e_w_i_t_h_a_n__o_b_je_c_t_iv_e__le_n_s_a_n_d__a_n_e_y_e_p_i_e_c_e_. _T_h_e_o_b_j_e_ct_ive
lens can magnify an object 104 times and the eyepiece can further magnify
an object 102 times. What is the maximum magnification on Ito’s
microscope?

_________________________________________________
_________________________________________________
_________________________________________________
3) J_o_y_t_y_p_e_d_a__to_t_a_l_o_f_1_0_7_w__o_r_d_s_in__1_0_4_d_a_y_s_._H_o_w__m__a_n_y_w_o__rd_s_did she type per
day?
____________________________________________________________
____________________________________________________________
____________________________________________________________
4) T_h_a_il_a_n_d_’s___p_o_p_u_la_t_io_n___in___2_0_1_7___is__a_b_o_u_t__6__.9__×_1__0_7_. __It_s__l_an__d__a_r_e_a_ is
approximately 5.1 × 105 square kilometers. Find the average population per
square kilometer.

_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________
_________________________________________________

5) The Mercury’s mass is about 0.055 times the Earth’s mass. The Earth’s mass
is about 5.98 × 1024 kilograms. Determine the mass of the Mercury in

kilograms.

_________________________________________ 15
_________________________________________
_________________________________________
_________________________________________
_________________________________________
_________________________________________
_________________________________________
_________________________________________

Mathematics Book 1

6) Mary invests $10,000 in private fund which pays 10% interest per annum
compounded annually. Find the amount of money after 2 years.
___________________________________________
___________________________________________
___________________________________________
___________________________________________
___________________________________________
___________________________________________
___________________________________________
___________________________________________
___________________________________________
___________________________________________

7) How much money will there be in an account at the end of 3 years if 1,000
Euros is deposited at 10% compounded annually?
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________

8) Julie accumulated 1,331,000 baht in an investment company. The investment
is at the rate of 10% per annum compounded annually in 3 years. How much
was her initial principal?
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________
____________________________________________________________

16 Mathematics Book 1

Summary

• Indices are used as the shorthand way of writing multiplications of the
same item several times.

• For any real number and for any positive integer ,
= … ( to times)

• Never multiply the base times the exponent.

base index, power or exponent

• Laws of Indices :

▪ = +

▪ = − , where ≠ 0


▪ ( ) =

▪ ( ) = , where ≠ 0 and ≠ 0

▪ ቀ ቁ = , where ≠ 0



▪ − =


▪ = , where ≠ 0

• Any quantity with a negative integer power is equal to the reciprocal of that
quantity with the corresponding positive exponent.

• A compound interest is the interest computed on the initial principal, which
also include the earned interest of previous time of a deposit or a loan.

= ( + )

When = accumulated amount (total amount)
= principal amount (initial amount)
= rate (percent in decimal form)
= time (in years)

Mathematics Book 1 17

Revision Exercise

Section A (Skills : 1 – 3)

1) Correct each wrong statement by changing the right side of the equation.

a) −32 = 9 = ___________________________________

b) 24 = 8 = ___________________________________

c) (3 )(3 ) = 3 = ___________________________________
= ___________________________________
d) (2 )3 = 3 = ___________________________________
2

e) 1 = 2−3
(−2)3

2) Find the values of the following indices.

a) 184 × 189 × 186 × 185 = ______________________________

b) 276 ÷ 278 ÷ 27−12 = ______________________________

c) (3 × 15)4 = ______________________________
= ______________________________
d) ቀ− 1 2 = ______________________________

17 ቁ

e) ሾ(56 4)0ሿ−1

3) Simplify and express the answers using only positive exponents.

a) 510×512×514 c) ൫ −3 2൯−3 × 3 2
511×513×515 4 −3 ( 5 −1)−5
__________________________ __________________________

__________________________ __________________________

__________________________ __________________________

__________________________ __________________________

b) 4 −9 −11 −13 d) 256−2×4−2×16
16 −7 −8 −6 24×2−5

__________________________ __________________________

__________________________ __________________________

__________________________ __________________________

__________________________ __________________________

18 Mathematics Book 1

Section B (Concepts : 4 – 8)

Read and analyze each question. Circle the letter of the correct answer.

4) Which of the following words is not 7) If = −3 and = −5, then find
related to indices? ÷ .
a) Exponent
b) Power a) 5
c) Index b) 2
d) Fraction c) −8
d) 0.6
5) Multiplying two expressions with the
same base mean to ___ their powers. 8) Which of the following statements is
a) subtract
b) add not true about laws of indices?
c) multiply
d) divide a) When you raise a power of a

6) All are true, except: power, add the exponents.
a) 30 + 12 = 13
b) 5−2 > 5−3 b) When dividing two expressions
c) ሾ(56)0ሿ−1 = 1
d) 45−1 = −45 with the same base, subtract

their exponents.

c) Any non-zero number raised to

the power of zero equal 1.

d) When you raise a quotient to a

power you raise both the

numerator and the denominator

to that power.

Section C (Word Problems : 9 – 20)

Read and analyze each question. Circle the letter of the correct answer.

9) A car travels at a speed 72 miles per 11) The object is about 30 × 1011 light

day. How long will the car travel in years far from the Earth. A light year
75 days? equals to 9.4 × 1012 kilometres.
a) 77 days
b) 77 miles Find the distance between this
c) 710 miles
d) 73 miles object and the Earth.
a) 2.82 × 1021 kilometres
e) None of the above b) 2.82 × 1023 kilometres
c) 2.82 × 1024 kilometres
10) There are 810 pieces of leaves in a d) 2.82 × 1025 kilometres
forest and there are 86 leaves in a
12) An atom has a diameter of
tree. How many trees are there? approximately 0.1 of a nanometres,
a) 816 trees or 0.0000000001 metres. Write the
b) 81.5 trees
c) 860 trees decimal as a power of 10.
d) 84 trees a) 10−1
b) 109
e) None of the above c) 10−9

Mathematics Book 1 19

d) 10−10 17) There are over 100 billion stars in our
e) None of the above galaxy, the Milky Way. Scientists
estimate there are about 100 billion
13) The fuel tank of Mr. Lee’s car has the galaxies in the universe. If every
dimensions 5 by 3 2 by 2 3. What galaxy has about 100 billion stars,
is the volume of the fuel tank? about how many stars are in the
(Volume = length × width × height) universe?
a) 5 5 6 a) 1 × 1019 stars
b) 5 5 5 b) 1 × 1020 stars
c) 6 5 5 c) 1 × 1021 stars
d) 6 7 3 d) 1 × 1022 stars

14) Bell’s rectangular room has a 18) The Hydra was a one-headed
capacity of 60 7 5 cubic unit. The monster but when it is cut off, 2
length is 3 7 units and a width of more heads grow in its place. If a
4 3 units. What is the height of her hero tried to conquer it by cutting
room? off all of its heads every day, how
a) 5 2 units many heads would the Hydra have
b) 12 3 units on the 7th day?
c) 5 units a) 14 heads
d) 20 2 units b) 128 heads
e) None of the above c) 2 heads
d) 49 heads
15) Lyn invests $500 in an account that
earns 5% interest compounded 19) The world’s smallest flowering plant
annually. Which expression gives the has a mass of 1.5 × 10−4 gram. At
balance in the account after 3 years? least 5 × 103 of these flowers could
a) 500 + (1+0.5) (1+0.5) (1+0.5) fit in a thimble. What is the mass of
b) 500(1+0.05) (1+0.05) (1+0.05) 5 × 103 flowers?
c) 500 × 0.05 × 0.05 × 0.05 a) 0.75 grams
d) 500 × 0.5 × 0.5 × 0.5 b) 7.5 grams
c) 0.075 grams
16) Akapon has a rare coin worth 350 d) 75 grams
Euros. Each decade, the coin value
increases by 10%. Which expression 20) An investment of 1,000B with 5%
gives the coin’s value, 5 decades interest paid each year and
from now? compounded. How much is the
a) 350 + (1 + 0.1)5 money after 2 years?
b) 350 + 0.15 a) 1,001 baht
c) 350(1 + 0.1)5 b) 2,100 baht
d) 350 × 0.15 c) 1,102.50 baht
d) 1,120.50 baht

20 Mathematics Book 1

UNIT 2

ALGEBRAIC EXPRESSIONS

BITS FROM THE PAST…

Until now you dealt with mathematics involving numbers. As we move
further we have to deal with unknown numbers. So to solve such type of problems
involving unknown numbers, a separate branch of mathematics was started. That
is ALGEBRA, the mathematics of unknowns.

Muhammad ibn Musa al-khwarizmi who lived from 800 –
847 CE, wrote a book titled “al-jabr” about arithmetic of
variables. The book was translated into Latin. Its title gave
Algebra its name.

Al-Khwarizmi called variables “shay”. Shay is an Arabic
name for things. The Spanish transliterated “shay” as “xay”,
which “x” was “shi” in Spanish. In time this word was
abbreviated as x.

Mathematical Terms monomial (เอกนาม)
multiplying (การคณู )
absolute value (ระยะหา่ งจากศนู ยค์ งึ จานวนนนั้ ) perform (ปฏิบตั ิการ)
adding (การบวก) polynomial (พหนุ าม)
acceptable (ยอมรบั ได)้ possible (เป็นไปได)้
algebraic expressions (นพิ จน)์ prerequisite (ท่จี าเป็นตอ้ งมี)
cancelling method (การตดั ทอน) quotient (ผลหาร)
coefficient (คา่ สมั ประสทิ ธิ)์ rearrange (จดั เรยี ง)
constant (คา่ คงตวั ) remainder (เศษ)
degree of polynomial (ระดบั พหนุ าม) subtracting (การลบ)
descending order (จากมากไปนอ้ ย) term (พจน)์
distributive property (สมบตั กิ ารกระจาย) variable (ตวั แปร)
dividend (ตวั ตงั้ )
divisor (ตวั หาร) Mathematics Book 1

21

1. Monomial

Monomial is a part of algebraic expression. Monomial is either a number, a

variable or the product of a number and one or more variables with whole number
powers. It cannot contain any addition or subtraction signs.

Study the monomial below:

Degree of monomial

Coefficient

Variable

• Variable is any letters in the alphabet representing unknowns.
• Degree of monomial is the sum of the indices of variables.
• Coefficient is the number beside the variable.

Example 1

Complete the table below. The first row is done for you.

Algebraic Monomial Not Coefficient Variable/s Sum of Degree of
expressions Monomial the monomials

5 −− powers 0

−2 0

− + 6
7
3

4 3



Solution

Algebraic Monomial Not Coefficient Variable/s Sum of the Degree of
expressions Monomial powers monomials
− −
5 00

−2 −2 1 1

22 Mathematics Book 1

Algebraic Monomial Not Coefficient Variable/s Sum of the Degree of
expressions Monomial powers monomials

− + 6 − , − −

7 1 , 7 + 1 8
3
3
4 3
4 , , 1+3 5
+1

− , , − −

Note: From example 1

• 5 is a constant monomial since there is no variable after the number.

• − + 6 is not a monomial since an addition sign is included.

• is not a monomial since it can be written as −1, and has a


negative power or index.

Exercise 2A

1) Tell whether each of the following expressions is a monomial or not a
monomial by filling in the blanks. If it’s not a monomial state your reason.

Algebraic Monomial Not a Reason
Expressions monomial

−10

3

1 2
2

5 +

2

−1.8 3

3 + 2 − 6

2ℎ2
−2
10 2
5

Mathematics Book 1 23

2) Simplify the given monomials, then fill in the blanks with the correct answers.

Monomial Coefficient Variable Degree

3

7 3

1
−2

(5 2 2)3

3 7
−5

3 4
−6

25 −6 −7
5 −9 −10

6 2

1 1
(3
3)2

−19 6 10 −3

7
9

(−2)4 2 11

(−3)−2( 3)4

2 (− 8
5)

24 Mathematics Book 1

2. Adding and Subtracting Monomials

To add or subtract two or more monomials, they have to be similar. Two
monomials are similar when they have the same variables with the same indices.

1.5 2 3 4 −2 2 3 4

2 3 4 ➢ similar variables and indices

To add two or more monomials:

➢ Add the coefficients
➢ Keep the variables

34 2 + 6 2 = (34 + 6) 2
= 40 2

−2 9 + 67 9 = (−2 + 67) 9
= 65 9

To subtract two or more monomials:

➢ Subtract the coefficients
➢ Keep the variables

34 2 − 6 2 = (34 − 6) 2
= 28 2

−2 9 − 67 9 = (−2 − 67) 9
= −69 9

Note Taking

• To add coefficients with same sign, keep the sign and add the absolute value of each
number. ( . . ∶ 2 + 3 = 5 ; (−2) + (−3) = −5)

• To add with different signs, keep the sign of the number with the biggest absolute value
and subtract the smallest absolute value from the biggest. ( . . ∶ −5 + 3 = −2)

• To subtract coefficients, add the opposite. ( . ∶ 2 − 3 = −1 ; 3 − 2 = 1 ;
−2 − 3 = −5 ; −2 − (−3) = −2 + 3 = 1 ; −5 − (−2) = −5 + 2 = −3)

Mathematics Book 1 25

Example 2

Add or subtract the following sets of monomials if possible.

1) 18 5 + 8 5 4) −17 2 3 − 15 2 3
2) −5 2 + 19 2 5) 6 3 2 − 9 2 3
3) 17 5 6 7 + 13 6 7 5 6) 8 2 − 15 2

Solution

1) 18 5 + 8 5 = (18 + 8) 5
=

2) −5 2 + 19 2 = (−5 + 19) 2
=

3) 17 5 6 7 + 13 6 7 5 ( not possible to add since the powers of
each variable is not the same )
4) −17 2 3 − 15 2 3 =
= (−17 − 15) 2 3
−
5) 6 3 2 − 9 2 3 =
= 6 2 3 − 9 2 3 (rearrange the variables)
= (6 − 9) 2 3
−
6) 8 2 − 15 2
( not possible to subtract since the numbers
of variables are not equal)

Example 3

Perform the indicated operations.

1) (−7 3 + 9 3 + (− 3)
2) 25 4 + (−19 4 ) + (−16 4 ) + 17 4
3) −3 − (−4 ) − (−8 )
4) 15 3 + (−7 3) − (−9 3 − 5 3)

Solution = ሾ−7 + 9 + (−1)ሿ 3
= ሾ−7 + 9 − 1ሿ 3
1) (−7 3 + 9 3 + (− 3) = ሾ2 − 1ሿ 3
= 1 3
=

26 Mathematics Book 1

2) 25 4 + (−19 4 ) + (−16 4 ) + 17 4
= ሾ25 + (−19) + (−16) + 17ሿ 4
= ሾ25 − 19 − 16 + 17ሿ 4
= ሾ6 − 16 + 17ሿ 4
= ሾ−10 + 17ሿ 4

=

3) −3 − (−4 ) − (−8 ) = ሾ−3 − (−4) − (−8)ሿ
= ሾ−3 + 4 + 8ሿ
= ሾ1 + 8ሿ
=

4) 15 3 + (−7 3) − (−9 3 − 5 3)

= ሾ15 + (−7) − (−9 − 5)ሿ 3
= ሾ15 − 7 + 9 + 5ሿ 3
= ሾ8 + 9 + 5ሿ 3
= ሾ17 + 5ሿ 3
=

Exercise 2B

1) Add the following algebraic expressions if possible.

a) + f) 5 2 + (−5 2)
____________________________ ____________________________

____________________________ ____________________________

____________________________ ____________________________

b) −15 5 + (−9 5 ) g) −2 + 17 + (−9 )
____________________________ ____________________________

____________________________ ____________________________

____________________________ ____________________________

c) −19 2 3 + 26 3 2 h) 5 9 + 4 9 + (−8 9) + 13 9
____________________________ ____________________________

____________________________ ____________________________

____________________________ ____________________________

d) 9 + (− ) i) 7 + 21 + (−19 ) + (−15 )
____________________________ ____________________________

____________________________ ____________________________

____________________________ ____________________________

e) 11 ℎ3 2 + 19 ℎ3 2 j) 2 + ቀ− 8 ቁ + ቀ− 6 ቁ +
55 5
55
____________________________
____________________________

____________________________ ____________________________

____________________________ ____________________________

Mathematics Book 1 27

2) Subtract the following algebraic expressions if possible.

a) − f) 5 2 − (−5 2 )
____________________________ ____________________________

____________________________ ____________________________

____________________________ ____________________________

b) −15 5 − (−9 5 ) g) −2 − 17 − (−9 )
____________________________ ____________________________

____________________________ ____________________________

____________________________ ____________________________

c) −19 3 2 − 26 3 2 h) 5 9 − 4 9 + (−8 9) − 13 9
____________________________ ____________________________

____________________________ ____________________________

____________________________ ____________________________

d) 9 2 − (− ) i) 7 − 21 − (−19 ) − (−15 )
____________________________ ____________________________

____________________________ ____________________________

____________________________ ____________________________

e) 11 ℎ3 2 − 19 ℎ3 2 j) 2 + ቀ− 8 ቁ − 6 +
55 5 5 5
____________________________ ____________________________

____________________________ ____________________________

____________________________ ____________________________

3) Perform the indicated operations.

a) _5_ _ _+__7_ _ _ _−__3_ _ _ _____________ e) (_−__1_6_ __−_1__1_ _)_−__(_−_5_ _ _+__9_ _ )____
____________________________
____________________________ ____________________________
____________________________
____________________________
f) (_7_ _ 2__−__9_ _2_)_+__(_1_5_ _ 2__−__1_3_ _2_)___
____________________________ ____________________________
____________________________
b) −__2_0_ _ 3_ _ _6_+__5_ _3_ _ 6__−__2_ _3_ _6_____ ____________________________

____________________________ h) 2 + (−8 2) + 6 − 5 2
____________________________
____________________________ ____________________________
____________________________
____________________________ ____________________________

c) 18ℎ3 4 − (−5ℎ3 4) + 10ℎ3 4 i) 7 − 5 + (−5 ) − 6 − 2
____________________________ ____________________________
____________________________
____________________________ ____________________________
____________________________
____________________________

____________________________

d) 6 2 + 6 2 − ቀ− 8 2 ቁ
________7__________7__________

____________________________

____________________________

____________________________

28 Mathematics Book 1

3. Multiplying Monomial

In chapter 1, you’ve learned about the Laws of Indices. These laws are
prerequisite to continue further about multiplying sets of monomials.

PREREQUISITE KNOWLEDGE:

× = + (Multiplying Powers with the Same Bases)

( + ) = + (Distributive Property)

negative × negative = positive (Rules of multiplying integers)

positive × negative = negative

When multiplying monomial by another monomial, first MULTIPLY the
coefficients and then ADD the powers of the same variables.

Example 4

Find the product of the following expressions.

a) 4(3 ) c) (−5 2 )(−3 2)

b) 2 (−5 ) d) 2(9 2 3)(− 5 2)

Solution = (4 × 3)
=
a) 4(3 )

b) 2 (−5 ) = ሾ2 × (−5)ሿ( × )
= −

c) (−5 2 )(−3 2) = ሾ(−5)(−3)ሿ( 2 • )( • 2)
= 15( 2+1)( 1+2)
=

d) 2(9 2 3)(− 5 2) = ሾ2 × 9 × (−1)ሿ( 2 • 5)( • )( 3 • 2)
= ሾ−18ሿ( 2+5)( 1+1)( 3+2)
= ሾ−18ሿ( 7)( 2)( 5)
= −

Mathematics Book 1 29

Multiply the following expressions. Note Taking
a) 2(3 + 12)
b) (7 )(3 3 − 4 ) Multiplying monomial by a
c) (−2 2 )(3 3 + 6 2 7 − 2 ) polynomial is to multiply each
term of the polynomial by the
monomial using distributive

property. Polynomial is the
word for two or more

monomials separated by one
or more of the four operations

(+,×,÷, −).

a) 2(3 + 12) = 2(3 ) + 2(12)
b) (7 )(3 3 − 4 ) = +

= (7 )(3 3) + (7 )(−4 )
= (7 • 3)( 1+3) + (7 • −4)( )
= −

c) (−2 2 )(3 3 + 6 2 7 − 2 3)
= −2 2 (3 3) + (−2 2 )(6 2 7) + (−2 2 )(−2 3)
= −6( 2+1 1+3) − 12( 2+2 1+7) + 4( 2+1 1+3)
= −6 3 4 − 12 4 8 + 4 3 4
= − −

Exercise 2C e) 8 (−5 5)

1) Find the product. _______________________________
_______________________________
a) 6(7 )
f) 0_(_1__4_5 2 21)
_______________________________
_______________________________ _______________________________
_______________________________
b) (_ _ _ _)_(_− )
g) 7__(_2_ _ _ )(−3 2 5)
_______________________________
_______________________________ _______________________________
_______________________________
c) 3__ _ _(_5_ )
h) _2_0_ _ _3_ 2 (−6 )
_______________________________
_______________________________ _______________________________
_______________________________
d) (_−__8_ _ _)(8 4) ______

_______________________________
_______________________________
______

30 Mathematics Book 1

2) Multiply the following expressions.

a) 3(2 − 5) = _________________________________
b) ( 2 − 7 ) _________________________________
c) 4 ( 3 2 + 7 ) _________________________________

= _________________________________
_________________________________
_________________________________

= _________________________________
_________________________________
_________________________________

d) (− )(− + 4) = _________________________________
_________________________________
_________________________________

e) (2 3 + 8 )(− ) = _________________________________
_________________________________
_________________________________

f) 4 (5 2 + 6 2) = _________________________________
_________________________________
_________________________________

g) (−3 )(5 + 3) = _________________________________
_________________________________
_________________________________

h) (−4 3)(5 3 − 6 + 1) = _________________________________
_________________________________
_________________________________

i) (− )( 2 + 3 − 4) = _________________________________

_________________________________

_________________________________

j) − ( 2 + + 2 = _________________________________

_________________________________

_________________________________

k) −19 3 2( 4 + 2 2 − 3 4) = _________________________________

_________________________________

_________________________________

l) ቀ− 2 3ቁ (18 3 3) = _________________________________
_________________________________
3

_________________________________

_________________________________

Mathematics Book 1 31

4. Dividing Monomial

To divide monomial by a monomial, first DIVIDE the coefficients and then
SUBTRACT the powers of the same variables.

PREREQUISITE KNOWLEDGE:

÷ = − (Dividing Powers with the Same Bases)

Negative ÷ Negative = Positive (Rules of dividing integers)

Positive ÷ Negative = Negative

Example 6

Find the quotient.
a) 25 4 ÷ 5
b) −8 5 ÷ 2 3
c) −49 7 5 ÷ (−7 2)

Solution

a) 25 4 ÷ 5 = 25 4 = 25 5 4) =
5
51 (

b) −8 5 ÷ 2 3 = − 8 5 = − 8 4 = −
2 3
21 ( 5−3)

c) −49 7 5 ÷ (−7 2) −49 7 5

= −7 2

= − 49 7

71 ( 7−1 5−2)

=

Example 7

Divide and simplify the expressions below.

a) (3 + 6 2) ÷ 3 65 3 2+30 2 5−45 4 4

c) 5 2

b) (24 7 − 8 4) ÷ 4 2 3 3 4−8 5 3−48 4 6

d) 4 3 3

32 Mathematics Book 1

Dividing a polynomial by a monomial is to divide each term of the

polynomial by a monomial.

+ + = + + ( ≠ 0)



Solution = 3 +6 2 = 1 3 2 6 2
3
a) (3 + 6 2) ÷ 3 +
31 31
= +

b) (24 7 − 8 4) ÷ 4 2 24 7−8 4 624 7 28 4
= 4 2 = 14 2 − 14 2

= −

65 3 2+30 2 5−45 4 4 1365 3 2 6 30 2 5 −94155 24 4
15 2 15 2
c) 5 2 = +
=
+ −

3 3 4−8 5 3−48 4 6 = 3 3 4 −2184 35 33 −124148 34 36
4 3 3
d) 4 3 3

= − −



Exercise 2D

1) Divide the following pairs of monomials.

a) 27 2 ÷ 3 = ______________________________________

b) 27 2 ÷ 3 ______________________________________

c) 13 7 ÷ (− 2) = ______________________________________

d) (56 4 4) ÷ 7 3 ______________________________________

e) 42 6 6 6 ÷ (−7 3 2 ) = ______________________________________

f) − 11 3 4 ÷ 13 ______________________________________
13
12 = ______________________________________

______________________________________

= ______________________________________

______________________________________

= ______________________________________

______________________________________
______________________________________

Mathematics Book 1 33

2) Find the quotient.

a) (6 − 3) ÷ 4 = _____________________________________

_____________________________________

b) (18 3 + 12 ) ÷ 6 _____________________________________

= _____________________________________

_____________________________________

c) (81 3 − 27 5) ÷ 9 3 _____________________________________

= _____________________________________

_____________________________________

_____________________________________

d) (−18 4 2 + 36 2 4) ÷ (−4 )

_____________________________________

_____________________________________

30 5−24 4−20 2 _____________________________________

e) 4 2 = _____________________________________

_____________________________________

_____________________________________

_____________________________________

3 5+2 6− 2 = _____________________________________

f) 2 _____________________________________
_____________________________________
_____________________________________

56 2 +96 2 = _____________________________________

g) 8 _____________________________________
_____________________________________
_____________________________________

5 4 2+9 3 3−3 2 4 = _____________________________________

h) −3 2 _____________________________________
_____________________________________
_____________________________________

27 5 3−18 4 4−36 5 6

i) −9 4 3 = _____________________________________

_____________________________________
_____________________________________
_____________________________________

−9 3 2+3 5 = _____________________________________

j) − _____________________________________
_____________________________________
_____________________________________

34 Mathematics Book 1

5. Polynomial

Polynomial comes from the word “poly” which means “many” and “nomial”
which means “term”. It literally means many terms.

Algebraically, polynomial is an algebraic expression consisting of constants
and/or monomial or two or more monomials. It is separated by one or more of the
four operations (+,×,÷, −), except NOT DIVISION BY A VARIABLE. It means only
POSITIVE INTEGER POWERS of variables are acceptable.

+ − constant

2 monomials

The example above has 3 terms: 2 monomials and a constant.

The degree of a polynomial is the highest power of monomial in a given
polynomial. From the same example, the degree of polynomial is 3 since 2 has
powers of 2 and 1 respectively and its sum, 2 + 1 = 3.

There are special names for polynomials with one, two or three monomials:

+ − +

Monomial Binomial Trinomial
(1 term) (2 terms) (3 terms)

The proper form of writing a polynomial is to put the monomial with the
highest degree first. Arrange the terms in descending powers. Make sure the
polynomial is simplified before arranging the terms.

Example 8

Determine whether each of the following algebraic expressions is a
polynomial or not. If it’s not, state the reason.

1) 5 4) 5 −1 +

2) 3 2 − 24 2

4 5) − 1
3
3) 2 2

Mathematics Book 1 35

Solution

a) 5 is a polynomial. It is just a constant.

b) 3 2 − 24 is a polynomial.

c) 4 2 2 is a polynomial
3

d) 5 −1 + is not a polynomial because it contains a negative power.

2

e) − 1 is not a polynomial because dividing by a variable is not allowed.

Example 9

Simplify and write in standard form. Then, determine the degree of each
polynomial and name the polynomial according to its number of term/s.
a) −5 2 + 7 2 + 9 2 − 3 2
b) 12 3 2 + 7 2 − 12 3 2 − 9 2 + 4 − 5

Solution

a) −5 2 + 7 2 + 9 2 − 3 2
= (7 2 + 9 2 ) + (−5 2 − 3 2)
= 16 2 + (−8 2)
= 16 2 − 8 2

Therefore, it can be simplified to − , so it is a binomial with
the degree of 3.

b) 12 3 2 + 7 2 − 12 3 2 − 9 2 + 4 − 5
= (12 3 3 − 12 3 3) + (7 2 − 9 2) + 4 − 5
= 0 + (−2 2) + 4 − 5
= 4 − 2 2 − 5

Therefore, it can be simplified to − − , so it is a trinomial with
the degree of 4.

36 Mathematics Book 1

Exercise 2E

1) Simplify the following polynomials and then arrange in descending powers.

a) 5 3 − 4 2 + 9 3 − 5 4 − 7 2 + 6 ________________________________
______________________________________________________________
______________________________________________________________

b) 4 4 + 2 − 4 4 + 1 _____________________________________________
______________________________________________________________
______________________________________________________________

c) 2 3 − 4 3 − 5 3 + 7 3 ___________________________________
______________________________________________________________
______________________________________________________________

d) 11 2 − 4 − 2 − 5 2 − 4 + 2 2 − 1 ________________________
______________________________________________________________
______________________________________________________________

e) 9 3 + 6 4 − 3 3 + 9 2 − _____________________________________
______________________________________________________________
______________________________________________________________

2) Complete the table below. The first row is done for you.

Algebraic expressions Polynomial Not a Number Degree of polynomial
polynomial of terms
5 − 3 4 + 3 − 6th degree(1 + 5 = 6)
1 4

−2
− 2

− − 9 2 − 4 3
4 − 5
3 +
+
+ 6
+ 6

6 + 2 3 − + 3



1 2 + 3 7 − 2
2

Mathematics Book 1 37

6. Multiplying Polynomials

There are two ways of multiplying polynomials by a polynomial:
➢ Multiplying polynomials in horizontal form,
It means to use the distributive property to multiply each term in the
first polynomial by each term in the second polynomial.
➢ Multiplying polynomials in vertical form,
Line up the polynomials as you would for numerical multiplication. Be
careful of the signs. If a term is missing, replacing it with 0 may help
you to line up the terms correctly.

Multiply the following polynomials in horizontal form.
a) ( + 1)(2 − 3)
b) (2 2 − 5)(− 3 + 2 − 1)

a) ( + 1)(2 − 3) = ( )(2 ) + ( )(−3) + 1(2 ) + 1(−3)
= 2 2 − 3 + 2 − 3
= − −

b) (2 2 − 5)(− 3 + 2 − 1)

= 2 2(− 3) + 2 2(2 ) + 2 2(−1) + (−5)(− 3) + (−5)(2 ) + (−5)(−1)
= −2 5 + 4 3 − 2 2 + 5 3 − 10 + 5
= − + − − +

Find the product of the following polynomials in vertical form.
a) (3 + 1)(2 2 − 3 )
b) (2 2 − 5 + 1)( 2 + 2 − 1)

38 Mathematics Book 1

Solution

a) × 3 + 1
2 2 −
+ 3
6 3 − 2 2 3 (2 2)(3 + 1)
+ − 9 2 − 3 (−3 )(3 + 1)
7 2 −
6 3

Therefore, the product is − − .

b) × 2 2 − 5 + 1
2 + 2 − 1

2 4 − 5 3 + 2 ( 2)(2 2 − 5 + 1)
(2 )(2 2 − 5 + 1)
+ 4 3 − 10 2 + 2 (−1)(2 2 − 5 + 1)

− 2 2 + 5 − 1

2 4 − 3 − 11 2 + 7 − 1

Therefore, the product is − − + − .

Exercise 2F

1) Find the product in horizontal form.

a) ( + 7)(3 + 9) = _____________________________________________
_____________________________________________________________
_____________________________________________________________

b) (6 + 2)2 = _________________________________________________
_____________________________________________________________
_____________________________________________________________

c) ( − 5)( 2 − 2 + 1) = ________________________________________
_____________________________________________________________
_____________________________________________________________

d) (7 − 2)(− 2 + 4 − 6) = ______________________________________
_____________________________________________________________
_____________________________________________________________

e) ( 2 − 1)(5 2 + 2 − 8) = ______________________________________
_____________________________________________________________
_____________________________________________________________
Mathematics Book 1 39

2) Find the product in vertical form.

a) (2 + 5)(3 2 − 7)
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

b) (−9 + 5)(4 2 − 12)
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

c) (3 2 + 1)( 2 + )
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

d) (4 2 − 7 − 9)( 2 + 6 + 5)
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

e) (−3 − 2 − 5 )(−3 − 2 − 5 )
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

40 Mathematics Book 1