Manhattan GMAT Strategy Guide Supplement _ Foundations of GMAT Math

Exponents & Roots

The negative of equals the negative of thewhich negative
three squared
quantity three squared, equals nine.

To calculate - 3 2, square the 3 before you multiply by negative one (-1). If you want to square the nega­
tive sign, throw parentheses around -3.

(-3)2 =9

The square of equals nine,

negative three

In (—3)2, the negative sign and the three are both inside the parentheses, so they both get squared. If you
say “negative three squared,” you probably mean (-3)2, but someone listening might write down - 3 2, so
say “the square of negative three” instead.

The powers o f—1 alternate between 1 and —1. Even powers o f—1 are always 1, while odd powers o f—1
are always -1.

(-1)1 = -1 = -1
(-1)2 =-lx-l =1
(-1)3 = —1 x —1x —1 = -1
(-1)4 = —1 x —1X—1 x —1 =1

Negative numbers raised to an even power are always positive. Negative numbers raised to an odd num­
ber are always negative.

(Negative)6™ = Positive (Negative)odd= Negative

A positive base raised to any power is always positive, because positive times positive is positive— no
matter how many times you multiply.

Since an even exponent gives you a positive result for both a positive and a negative base, an even expo­
nent can hide the sign of the base. Consider this equation:

x2= 16

In Chapter 6, “Equations,” we will cover in more depth how to solve an equation such as this one. For
now, notice that two numbers for x would make the equation true:

42= 16 (-4)2= 16

The value of x could be either 4 or —4. Always be careful when dealing with even exponents in equa­
tions. Look for more than one possible solution.

MANHATTAN

GMAT

Exponents & Roots

Check Your Skills
1. Which is greater, - 5 8or (-5 )8?

Answers can befound on page 131.

Multiply Termswith Same Base: Add the Exponents

Imagine that you multiply together a string of five as. Now multiply a second string of three a s togeth­
er. Finally, because you love multiplication, go ahead and multiply the two strings together. How many
a s do you end up with?

Write it all out longhand:
(a x a x a x a x a) x (a x a x a) - a x a x a x a x a x a x a x a

Now use exponential notation:

a5 x a5 = a8
a to the fifth times a to the third equals a to the eighth.

What happens to the exponents 5 and 3? They add up: 5 + 3 = 8. This works because we only have as
in the equation. The two terms on the left {a5and a3) have the same base (a), so we have eight as on
each side of the equation.

When you multiply exponential terms that have the same base, add the exponents.

Treat any term without an exponent as if it had an exponent of 1.

y ( y 6) = y x y 6 = y l X y 6 = y 1+6 = f

Adding exponents works with numbers in the base, even weird numbers such as n. You just have to
make sure that the bases are the same.

53 X 56 = 59 7TX7C2 = X5

The rule also works with variables in the exponent.

23x 2Z= 23+z 6(6*) = 61x 6 x= 6l+x = 6X+1

Check Your Skills

Simplify the following expressions.

2. b5x b7=
3 .(x 3)(x*) =

Answers can befound on page 131.

MANHATTAN

GMAT

Exponents & Roots Chapter 3

Divide Terms with Same Base: Subtract the Exponents

Now divide a string of five as by a string of three as. Again, these are strings of multiplied as. What is
the result?

-a--x--a--x---a--x--a--x---a = --a-X---f-i-X--f-l-X---f-l-X--a- —a x a
axaxa flXfiXfl

In exponential notation, you have this: — = a 2
a

What happens to the exponents? You subtract the bottom exponent from the top exponent. 5 - 3 = 2.

When you divide exponential terms that have the same base, subtract the exponents.

This rule works the same for numbers as for variables, either in the base or in the exponent.

M6 y

—2r1Dr = 216-13 = 23 = 8 ^ 2=. ^ - 2

X

Again, treat any term without an exponent as if it had an exponent of 1.

r9 r9
J _ _ J _ _ rS
ffJ

Just always make sure that the bases are the same.

Here s the rule book so far.

If you... Then you... Like this:
Add the exponents a1 x a5= o ’
Multiply exponential terms
that have the same base

Divide exponential terms Subtract the exponents o’ 2
that have the same base ~&S = a

Check Your Skills Answers can befound on page 131.

Simplify the following expressions.

4 .y4

d8

5.

M A N H A T T A N 109
GMAT

Chapter 3 Exponents & Roots

Pretty Much Anything to the Zeroth Power: One

Divide a string of five ds by a string of five as. As before, each string is internally multiplied. What do
you get?

Using longhand, you get 1.

a x a x a x a x a __4 X4 X4 XfiX4
a x a x a x a x a 4 x 4 x 41x 4 x 4

Using the exponent subtraction rule, you get a0.

J>
—a —a——a - ^ 5 ~ 5 -
a

So <2° must equal 1. That’s true for practically any value of a.

1° = 1 6.2° = 1 (-4)° = 1

The only value of a that doesn’t work is 0 itself. The expression 0° is undefined. Notice that the argu­
ment above required us to divide by a. Since you can’t divide by 0, you can’t raise 0 to the 0th power
either. The GMAT will never ask you to do so.

For any nonzero value of a , we can say that a 0= 1.

Now we can extend the powers of 2 to include 2°.

Powers of 2
2°= 1
21= 2
22= 4
23= 8
24= 16

The pattern should make sense. Each power of 2 is 2 times the previous power of 2.

Negative Power: One Over a Positive Power

What happens if you divide a string of three a s by a string of five as?

Using longhand, you get a leftover a2 in the denominator of the fraction.

aXaXa 1 _1

a x a x a x a x a aXaX4X4X4 ax a a1

m M ANHATTAN

GMAT

Exponents & Roots Chapter 3

Using the exponent subtraction rule, you get a~2.
3
—a r = a 3-5 = a -2
a

So those two results must be equal. Something with a negative exponent is just “one over” that
same thing with a positive exponent.

a
a to the negative two equals one over a squared

In other words, a~2 is the reciprocal of a1. The reciprocal of 5 is “one over” 5, or —. You can also think

of reciprocals this way: something times its reciprocal always equals 1. ^

5 x■-^ = l -a22^x ^1 _=! l a22wxa-~22 = a22-2-2 = ao° = 1
5 a

Now we can extend the powers of 2 to include negative exponents.

Powers of 2
2-} = -V = - =0.125

28

2"2= ^ = - =0.25
24

2_1 = — = 0.5

2

2° = 1
21 = 2
22= 4
23= 8
24 = 16

The pattern should still make sense. Each power of 2 is 2 times the previous power of 2.

The rules we’ve seen so far work the same for negative exponents.

5-3 x 5~6= 5"3+(-6) = 5~9

* L = xK -* > = x*

X

Negative exponents are tricky, so it can be useful to rewrite them using positive exponents. A negative
exponent in a term on top of a fraction becomes positive when you move the term to the bottom.

M AN HA TTA N
GMAT

Chapter 3 Exponents & Roots

5x 5

J/ 3 ” X 2 ) 3

Here, we moved x"2from the numerator to the denominator and switched the sign of the exponent from
- 2 to 2. Everything else stays the same.

Likewise, a negative exponent in the bottom of a fraction becomes positive when the term moves to the
top.

3 3z4
Z— 4 U) w22

Here, we moved z~4 from the denominator to the numerator and switched the sign of the exponent from
- 4 to 4.

If you move the entire denominator, leave a 1 behind.

1 1Xz’ 4
Z- 4 ~ 1-i ~ Z

The same is true for a numerator.

w5 1
2 2w

Don’t confuse the sign of the base with the sign of the exponent. A positive base raised to a negative
exponent stays positive.

3-3 = — = —
33 27

A negative base follows the same rules as before. Odd powers of a negative base produce negative num­
bers.

(-4)-3= 1 1 1
(-4) -64 64

Even powers of a negative base produce positive numbers.
1
•= ( - 6 ) = 36

(-6 )-

112 MANHATTAN

GMAT

Exponents & Roots Chapter 3

Here are additional rules for the rule book.

If you... Then you... Like this:

Raise anything to the zeroth Get one a°= 1
power (besides zero itself)

Raise anything to a negative Get one over that same thing to -2 _ 1

power the corresponding positive power a/72

Move a term from top to Switch the sign of the exponent 2a 2 _ 2
bottom of a fraction 3 3a2
(or vice versa)

Check Your Skills

Simplify the following expressions.

6. 2"3

Answers can befound on page 131.

ApplyTwo Exponents: Multiply the Exponents

How do you simplify this expression?

(*2)4

Use the definition of exponents. First you square a. Then you multiply four separate a2 terms together.
In longhand:

(a2)4 = a2 x a2 x a2 x a2= a2+2+2+2= a8

What happens to the exponents 2 and 4? You multiply them: 2 x 4 = 8. On each side, you have eight ds
multiplied together.

When you raise something that already has an exponent to another power, multiply the two
exponents together.

Always keep these two cases straight.

Addition rule: a2x a4= a2+4= a6

Multiplication rule: (a2)4= a2x4 = a8

MANHATTAN 113
GMAT

Chapter 3 Exponents & Roots

To add exponents, you should see two bases, as in x a4. To multiply exponents, you should be apply­
ing two exponents, one after the other, to just one base: (a2)4.

The “apply two exponents” rule works perfectly with negative exponents as well.
(*-3)5= *-3x5= x -15

(4~2)~3 = 4~2x~3 = 4 6

If you... Then you... Like this:
Multiply the powers
Raise something to two 00
successive powers
II

s

Put it all together. Now you can handle this expression:

x-} (x2)4

First, simplify (x2)4.
(x2)4= x2*4= x8

The fraction now reads:

-3 8

Now follow the rules for multiplying and dividing terms that have the same base. That is, you add and
subtract the exponents:

If you have different bases that are numbers, try breaking the bases down to prime factors. You might
discover that you can express everything in terms of one base.

22 x 43 x 16 =

(A) 26
(B) 212
(C) 218

114 MANHATTAN
GMAT

Exponents & Roots Chapter 3

The efficient way to attack this problem is to break down 4 and 16 into prime factors. Both 4 and 16 are
powers of 2, so we have:

4 = 22 and 16 = 24
Everything can now be expressed with 2 as the base.

22 x 43 x 16 = 22 x (22)3x 24
= 22x 26 x 24
_ 22+6+4

= 212
The correct answer is (B).

Check Your Skills

Simplify the following expressions:

8. (x3)4

a ° ( °3) Answers can befound on page 131.

Apply an Exponent to a Product: Applythe Exponent to Each Factor

Consider this expression:

(*# 115
How can you rewrite this? Use the definition of exponents. You multiply three xy terms together.

(xy)3= xyx xyx xy
So you have three x’s multiplied together and threey s multiplied together. You can group these up sepa­
rately, because everything’s multiplied.

(xy)3= xy x xy x xy = x3)/3

When you apply an exponent to a product, apply the exponent to each factor.
This rule works with every kind of base and exponent weve seen so far.

(3x)4 = 34x4 = 81x4
(lU2?)x= U?2?x

MANHATTAN
GMAT

Chapter 3 Exponents & Roots

( 2 - y y 3= 2-2x- y x-3= 26y~6= GAy 6= ——

You do the same thing with division. In particular, if you raise an entire fraction to a power, you sepa­
rately apply the exponent to the numerator and to the denominator.

4 4 x 32 36

3”2
In ^ , the exponent applies only to the numerator (3). Respect PEMDAS, as always. Here’s more for

the rule book.

If you... Then you... Like this:IT
a^ \\4 aII
Apply an exponent Apply the exponent
to a product to each factor

Apply an exponent Apply the exponent
to an entire fraction separately to top and bottom

You can use this principle to write the prime factorization of big numbers without computing those
numbers directly.

What is the prime factorization of 183?

Don’t multiply out 18 x 18 x 18. Just figure out the prime factorization of 18 itself, then apply the rule
above.

18 = 2 x 9 = 2 x 32

183= (2 x 32)3= 23 x 36= 2336

Now simplify this harder example.

122 X 8 _
18

m MANHATTAN

GMAT

Exponents & Roots Chapter 3

First, break each base into its prime factors and substitute.

12 = 22 x 3 8 = 23 18 = 2 x 32

122 x 8 (22X3)2X23
2x3
18

Now apply the exponent on the parentheses.

(2 2 X 3)2 X 23 24x 32 x 23

2x3 2x3

Finally, combine the terms with 2 as their base. Remember that a 2 without a written exponent really
has an exponent of 1. Separately, combine the terms with 3 as their base.

2 X3 * 2 = 24+3_1 x 3 2- 2 = 26 x 3° = 26 x 1= 26 = 64
2x3

Occasionally, its faster not to break down all the way to primes. If you spot a larger common base, feel
free to use it. Try this example:

36^ =

64

You can simplify this expression by breaking 36 and 6 down to primes. But if you recognize that 36 =
62, then you can go much faster:

363 (62f 66 2

~7T~ ~ Tk = “TT = 6 = 3 6

One last point: be ready to rewrite *z3£3 as (<ab)3.

Consider 24 x 34. Here’s a way to see that 24 x 34 equals (2 x 3)4, or 64:

24 x 34= (2 x 2 x 2 x 2) x (3 x 3 x 3 x 3)

= (2 x 3) x (2 x 3) x (2 x 3) x (2 x 3) by regrouping

= (2 x 3)4 = 64

More often you need to change (a b f into a?b5, but occasionally it’s handy to go in reverse.

MANHATTAN 117

GMAT

Chapters Exponents & Roots

If you... Then you... Like this:
a3P = (al?)5
See two factors with the same Might regroup the factors
exponent as a product

Check Your Skills
Simplify the following expressions.

V/ Answers can befound on page 131.
^ 753X453

158

Add or Subtract Terms with the Same Base: Pull Out a Common Factor

Every case so far in this chapter has involved only multiplication and division. What if you are adding or
subtracting exponential terms?

Consider this example:

135+ 133=

Do not be tempted to add the exponents and get 138. That is the answer to a similar but different ques­
tion (namely, what is 135 x 133?). The right answer to a different question is always wrong.

Instead, look for a common factor and pull it out. Both 135and 133 are divisible by 133, so thats your
common factor. If necessary, rewrite 135as 133132first.

135+ 133= 133x 132+ 133= 133(132+ 1)

You could go further and rewrite 132 as 169. The right answer choice would possibly look like this:
133(170).

If we had xs instead of 13 s as bases, the factoring would work the same way.

X5 + X5 — X3 X X 1 + X5 = x ^ ix 1 + 1)

Try this example:
3« _ 37 _ 36 =

(A) 36(5)
(B) 36
(C) 3-5

118 MANHATTAN

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Exponents & Roots Chapter 3

All three terms (38, 37, and 36) are divisible by 36, so pull 36out of the expression:

38 _ 37 _ 36 = 36(32 _ 31 _ 30) = 36(9 _ 3 _ 1) = 36(5)

The correct answer is (A).

Now try to simplify this fraction:

34 + 3 5 + 3 6
13

Ignore the 13 on the bottom of the fraction for the moment. On the top, each term is divisible by 34.

34 + 3 5 + 3 6 34(3° + 31+ 32)
13 13

Continue to simplify the small powers of 3 in the parentheses:

34 + 3 5 + 36 _ 34(3° + 31 + 32) _ 34(l + 3 + 9) _ 34(13)

13 13 13 13

Now we can cancel the 13s on the top and bottom of the fraction.

34+ 35 + 36 _ 34(3° + 3‘ + 32) _ 34(1 + 3 + 9) _ 34(13)
= 34

13 13 13 13

If you don’t have the same bases in what you’re adding or subtracting, you can’t immediately factor. If
the bases are numbers, break them down to smaller factors and see whether you now have anything in
common.

46+ 206=

Again, don’t answer the wrong question. 46x 206= (4 x 20)6= 806, but that doesn’t answer this ques­
tion. We need to add 46and 206, not multiply them.

Since 4 is a factor of 20, rewrite 20 as 4 x 5 and apply the exponent to that product.

46 + 206= 46+ (4 x 5)6= 46+ 46x 56

Now pull out the common factor of 46.

46+ 206= 46+ (4 x 5)6= 46+ 46x 56= (1 + 56)46.

MANHATTAN 119

GMAT

Exponents & Roots

Thats as far as you’d reasonably go, given the size of 46and 56. Finally, try this one:
45+ 203 =

Start itthe same way as before. Rewrite 20 as 4 x 5 and apply the exponent.
45+ 203 = 45+ (4 x 5)3 = 45+ 43 x 53

Now, the common factor is only 43.
45+ 203 = 45+ (4 x 5)3 = 45+ 43 x 53 = 43 x 42+ 43 x 53 = 43(42+ 53)

Tlie result isn’t especially pretty, but it’s legitimate. Here’s the rule book:

If you... Then you... Like this:

Add or subtract terms with Pull out the common factor 23 + 25
the same base = 2 3(1 + 2 2)
Break down the bases and
Add or subtract terms with pull out the common factor 23 + 63
different bases = 2 3(1 + 3 3)

Check Your Skills

Simplify the following expression by factoring out a common term:

12. 55+ 54- 53

Answers can befound on page 131.

Roots: Opposite of Exponents

Squaring a number means raising it to the 2nd power (or multiplying it by itself). Square-rooting a
number undoes that process.

32 = 9and yfe) =3

Three is nine, and the square is three,

squared root of nine

If you square-root first, then square, you get back to the original number.

(VT6 )2 = VI6 x j l 6 = 16

The square of the equals the square root of which sixteen,
square root of sixteen sixteen times the equals
square root of sixteen,

MANHATTAN

GMAT

Exponents & Roots Chapter 3

If you square first, then square-root, you get back to the original number if the original number is posi­
tive.

V5x5

The square root of equals the square root of which five,

five squared five times five, equals

If the original number is negative, you just flip the sign, so you end up with a positive.

a/25 = 5

The square root of the equals the square root of which five,

square of negative five twenty-five, equals

In fact, square-rooting the square of something is just like taking the absolute value of that thing (see
Chapter 8).

If you... Then you... Like this:
Square a square root Get the original number
Or-H
II
(N

Square-root a square Get the absolute value of the Vio2= io
original number
7 ( - 10)2 = 1 0

Because 9 is the square of an integer (9 = 32), 9 is a perfect square and has a nice integer square root. In
contrast, 2 is not the square of an integer, so its square root is an ugly decimal, as we saw in Chapter 1.

Memorize the perfect squares on page 106 so you can take their square roots easily. Also memorize
these approximations:

72=1.4 V 3 -1.7

You can approximate the square root of a non-perfect square by looking at nearby perfect squares. The
square root of a bigger number is always bigger than the square root of a smaller number. Try this
example:

V70 is between what two integers?

Two nearby perfect squares are 64 and 81. \f64 = 8 and T = 9, so V70 must be between 8 and 9.
Its closer to 8, in fact, but you wont have to approximate the decimal part.

MANHATTAN 121
GMAT

Exponents & Roots

The square root of a number bigger than 1 is smaller than the original number.

J2< 2 \f2l < 21 VL3<1.3

However, the square root of a number between 1 and 0 is bigger than the original number.

y/0.5 >0.5 (y /0 5 *> 0 j)

In either case, the square root of a number is closer to 1 than the original number.

The square root of 1 is 1, since l 2= 1. Likewise, the square root of 0 is 0, since 02= 0.

VT=i Vo=o

You cannot take the square root of a negative number in GMAT world. What is inside the radical sign
must never be negative.

Likewise, the square root symbol never gives a negative result. This may seem strange. After all, both
52and (-5 )2equal 25, so shouldn’t the square root of 25 be either 5 or -5? No. Mathematicians like to
have symbols mean one thing.

>/25 = 5 and that’s that.

When you see the square root symbol on the GMAT, only consider the positive root.

In contrast, when you take the square root of both sides of an equation, you have to consider both posi­
tive and negative roots.

a; = V25 solution: x = 5

x2 = 25 solutions: x = 5 O R x = —5

Be careful with square roots of variable expressions. The expression must not be negative, or the square
root is illegal.

Check Your Skills

13. V27x V27 =

Answers can befound on page 131.

MANHATTAN

GMAT

Exponents & Roots Chapter 3

Square Root: Power of One Half

Consider this equation:
(9xf = 9

What is xi We can find x using tools we already have.
( r ) 2 =9
92* = 91

The exponents must be equal. So 2x = 1, or x = .

Now we know that ^9^ j = 9. We also know that (V9) = 9. So we can conclude that y/9 = 9^.
For expressions with positive bases, a square root is equivalent to an exponent of —.

2
Try to simplify this example:

1/7 “ =
You can approach the problem in either of two ways.
1. Rewrite the square root as an exponent of —, then apply the two-exponent rule (multiply exponents).

y jj22 = ^7 22 _ y2% _ y11

2. Rewrite whats inside the square root as a product of two equal factors. The square root is therefore
one of those factors.

722 = 7 n x 7 n

= 4 ? UX7U = 7 u

Notice that you get an exponent that is exactly h a lfo t the exponent inside the square root. This tells you
that a number such as I 11 is a perfect square: 722= (711)2- An integer raised to a positive even power is
always a perfect square.

MANHATTAN 123

GMAT

Exponents &Roots

Heres the rule book:

If you... Then you... Like this:

Take a square root of a Rewrite the square root as >/5ir = ( 5 n f
positive number raised = 56
an exponent of —, then
to a power multiply exponents

OR 7 5 ir = V 56x 5 6
Rewrite whats inside the = 56
root as a product of two

equal factors

J.
Avoid changing the square root to an exponent of 2 when you have variable expressions inside the radi­
cal, since the output depends on the sign of the variables.

Check Your Skills
14. Ifxis positive, Vx®" =

Answers can befound on page 131.

Cube Roots Undo Cubing

Cubing a number means raising it to the 3rdpower. Cube-rooting a number undoes that process.

43 =64 and y /6 4 = 4

Four is sixty-four, and the cube is four,
cubed root of

sixty-four

Many of the properties of square roots carry over to cube roots. You can approximate cube roots the
same way

766 is a little more than 4, but less than 5, because y/64 = 4 andyj125 - 5.

Like square-rooting, cube-rooting a positive number pushes it toward 1.

yjY7 <17 but ^ 0 T 7 > 0 . 1 7

The main difference in behavior between square roots and cube roots is that you can take the cube root
of a negative number. You wind up with a negative number.

yJ-64 = - 4 because ( - 4 ) 3 = - 6 4

MANHATTAN
GMAT

Exponents & Roots Chapter 3

As a fractional exponent, cube roots are equivalent to exponents of —, just as square roots are equiva- 1
lent to exponents of —1 . Going further, fourth roots are equivalent to exponents of —, an^d so on.

24

We now can deal with fractional exponents. Consider this example:

8^ =

21
Rewrite — as 2 X —, making two successive exponents. This is the same as squaring first, then cube-
rooting. 3 3

8^ = 8i>‘^ = ( 82)K = # ' = ^/64=4

Since you can rewrite —2 as —1 x 2 instead, you can take the cube root first and then square the result, if

you like. ^^

8* = 8^ = ^

If you... Then you... Like this:

Raise a number to a Apply two exponents— the 125% = (V l2 5 )2
fractional power numerator as is and the = 52= 25
denominator as a root, in
either order

Check Your Skills

15.642/3=
Answers can befound on page 131.

Multiply Square Roots: Multiply the Insides

Consider this example:
V8 x V 2 =

We saw before that 8*2* = (8 x 2)a. This principle holds true for fractional exponents as well.

>/8xV 2 = 8^ x 2 ^ = ( 8 x 2 )^ = V 8 x 2

MANHATTAN 125
GMAT

Chapter3 Exponents&Roots

In practice, we can usually skip the fractional exponents. When you multiply square roots, multiply
the insides.

V8xV2 =V8x2 =Vl6=4

This shortcut works for division too. When you divide square roots, divide the insides.

S V3

As long as you’re only multiplying and dividing, you can deal with more complicated expressions.

V l 5 x V l 2 115x12 r—
— 35— ■

If you... Then you... Like this:
Multiply square roots y fa x jb = 4ab
Multiply the insides, then
Divide square roots square-root tri
ii
Divide the insides, then
square-root

Check Your Skills

Simplify the following expressions.

16. V20xV5

V384
17‘ V^xV3

Answers can befound on page 131.

Simplify Square Roots: Factor Out Squares

What does this product equal?

a/6 x ^2 =

First, you multiply the insides:
y[6xj2= y/l2

,26 MANHATTAN

GMAT

Exponents & Roots

You might think that you’re done— after all, 12 is not a perfect square, so you won’t get an integer out
of V l2 . -s/l~2 is mathematically correct, but it will never be a correct answer on the GMAT, because it
can be simplified. That is, y[\2 can be written in terms of smaller roots.

Here’s how. 12 has a perfect square as a factor. Namely, 12 = 4 x 3. So plug in this product and separate
the result into two roots.

7l2= 74x3 =74x73

The point of this exercise is that 7 4 is nice and tidy: 7 4 = 2. So finish up:
712=74x3=74x73=2^

If the GMAT asks you for the value of 7 6 X 7 2 , then 2 73 will be the answer.
To simplify square roots, factor out squares.

If you... Then you... Like this:
V50=V25x 2
Have the square root of a Pull square factors out of
large number (or a root that the number under the = V25xV2
radical sign = 5^2
doesn’t match any
answer choices)

Sometimes you can spot the square factor, if you know your perfect squares.
7360 =

360 should remind you of 36, which is a perfect square. 360 = 36 x 10.
7360 =736x10= 736 x7T0=67l0

What if you don’t spot a perfect square? You can always break the number down to primes. This
method is longer but guaranteed.

Consider V l2 again. The prime factorization of 12 is 2 x 2 x 3, or 22x 3.

7l2=722x3 = 7 ^ x 73=273

Each pair of prime factors under the radical (2 x 2, or 22) turns into a single copy as it emerges (becom­
ing the 2 in 2yji ). In this exercise, it can be useful to write out the prime factorization without expo­
nents, so that you can spot the prime pairs quickly.

MANHATTAN

GMAT

Chapter 3 Exponents & Roots

Take V360 again. Say you don’t spot the perfect square factor (36). Write out the prime factorization
o f360.

360

/\
(2 ) 180

/\
( 2) 90

/\
9 10

/\ /\

© @© ©

360 = 2 x 2 x 2 x 3 x 3 x 5

Now pair off two 2’s and two 3’s, leaving an extra 2 and 5 under the radical.

V360 = > / 2 x 2 x 2 x 3 x 3 x 5 = 7 2 x 2 x 7 3 x 3 x ^ 2 x 5 = 2 x 3 x 7 2 x 5 = 6 > /l0

Check Your Skills
Simplify the following roots.

18. 7 %
19. 7 4 4 1

Answers can befound on page 131.

Add or Subtract Inside the Root: Pull Out Common Square Factors

Consider this example:

Don’t be fooled. You cannot break this root into + y [^ . You can only break upproducts, not
sums, inside the square root. For instance, this is correct:

V32x 4 2 = ^ x > / 4 I = 3 x 4 = 12
To evaluate yjd2 + 4 2 , follow PEMDAS under the radical, then take the square root.

V32 + 4 2 = V9 + 16 =a/25 = 5

128 HANHATTAN
GMAT

Exponents&Roots Chapter3

The same goes for subtraction.

V l32 - 5 2 = 7 1 6 9 - 2 5 = V l44 =12
O ften you have to just crunch the num bers if they’re small. However, when the num bers get large, the
G M A T will give you a necessary shortcut: factoring out squares.
You’ll need to find a square factor that is com m on to both terms under the radical. This square term
will probably have an exponent in it.

V310+ 3" =
First, consider 3 10 + 3 11 by itself. W h a t is the largest factor that the two terms in the sum have in com­
mon? 3 10. N ote that 3 11 = 3 10 x 3.

3 io + 311 = 310(1 + 3 ) = 3 >o(4)

After you’ve factored, the addition becomes simply 1 + 3 . Now plug back into the square root.

V 310+ 3 U = V 310(l + 3 )= V 3 10(4 )= V 3 IFx V 4

Since 3 10 = (35)2, ^ 3 l° = 3 5. Alternatively, you can apply the square root as an exponent o f —:

310V = ( 3 10 = 3 ^ = 3 5. A nd, of course, V 4 = 2.

V 310 + 3 U = %/3, 0 ( l + 3 ) = ^ / 3 I0 ( 4 ) = ^ x V 4 = 3 5 x 2
The answer m ight be in the form 3 5(2).

If you... Then y o u ... Like this:

Add or subtract Factor out a square V 4 14+ 4 I 6 = ^ / 4 ,4 ( l + 4 2)
underneath the square factor from the sum or = V 4 14 X yj\ + 16
= 4 7V l7
root symbol difference
V 6“+ 8 2 = V36 + 64
OR = V ioo = 10
Go ahead and crunch
the num bers as written,

i f they’re small

Check Your Skills

20. Vi 0s —104 =

Answers can befou n d on page 131.

M A N H A T T A N 129

GMAT

Exponents & Roots Chapter 3

Check Your Skills Answer Key:

1. (—5)8: —58= —1 x 58, and is thus negative. (—5)8will be positive.
2. P * b7= b(i+7) = b n
3. (x5)(x4) = x i3+4> = x 7

5. ? - = d * - 7) = d
d1

6. 2-} = -
8

1
7. —r = 33 = 27

3
8. ^ x 4 = jc12

9. -= a 15-0-9 = U6

10. x y \2 2x2 1x2 4 2
* 7 *y 426

-3X2 = — - = * y z

,, 753 x 453 _ ( 3 x 52)3 x (32 x 5)3 _ (33 x 5 6)x (3 6 x 5 3) _ 39 x 5 9 _ , wC_ , e

15 (3 x 5) (3 x 5 ) 3 x5

12. 53(52 + 5 - 1)

13. 27: any square root times itself equals the number inside.

14. yfx^ = yjx* x x 3 = x 3. Since x is positive, x3 is positive too.

15. 642/3 = (7 6 4 )2 = (4)2 = l6

16. ^ 2 0 x 7 5 = V 20x5 = VlOO =10

n m m m r n . = j t i mt
72x73 v2x3 V 6

18. 7 % = 7 3 x 2 x 2 x 2 x 2 x 2 = 7 l 6 x 7 3 x 2 = 4 7 6

19. 7 4 4 1 ^ ^ / 3 x 3 x 7 x 7 = 7 3 x 3 x 7 7 x 7 = 3 x 7 = 21

20. 7 l 0 5 —104 = V l04( 1 0 - l) = 1027 9 = 1 0 0 x 3 = 300

MANHATTAN 131

GMAT

Exponents & Roots Chapter 3

Chapter Review: Drill Sets

Drill 1

Simplify the following expressions by combining like terms. If the base is a number, leave the answer in
exponential form (i.e. 23, not 8).

1. Xs x x3=
2. 76x 79=

-b
4. (o3)2=

5 .4-2 x 45=

6 .^ 1 =
(-3)2

7. (32)-3 =

8 .121-
11*

9. x2x x3x x 5=
10. (52)* =

Drill 2

Simplify the following expressions by combining like terms. If the base is a number, leave the answer in
exponential form (i.e. 23, not 8).

1 1 .34X 32X 3 =
■J2 x sx x 6

x2
56 x 5 4x
13.

54
14.y7x y8x y 6=

x4
15

16. z~—8 —j — =
32xx 3 6x

17- ^ i T - =
18. (x2)6x x3=
19.(z6)xX 23x=

2 0 .^ =
(5 )

MANHATTAN 133
GMAT

Chapter 3 Exponents & Roots

Drill 3

Follow the directions for each question.

21. Compute the sum. 273+ 9 2 +-^- = ?
9

22. Which of the following has the lowest value?
(A) (-3 )4
(B) - 3 3
(C) (-3 )-3
(D) (-2 )3
(E) 2-6
/^3 2

23. Compute the sum. 6 3- - + 8 3 =?
6

24. Which of the following is equal to
v5 ,

\3

<A) - f t

(B)
v5 /

(C) 2^3
v ’5 ,

(E) ' 5 ?

134 MANHATTAN

GMAT

Exponents & Roots Chapter 3

25. Which of the following has a value less than 1? (Select all th at apply)

(E) (-4)3

Drill 4

Simplify the following expressions by finding common bases.

2 6 .83 x 26

27.492x 77
28.254x 1253
2 9 .9~2x 272
3 0 .2~7 x 82

Drill 5

Simplify the following expressions by pulling out as many common factors as possible.

31.63+ 33=

(A) 3s (B) 39 (C) 2(33)

32.813+ 274=

(A) 37(2) (B) 312(2) (C) 314

3 3 .152- 5 2=

(A) 52(2) (B) 5223 (C) 5232

34.43+ 43+ 43+ 43+ 32+ 32+ 32=

(A)44+ 33 (B)4,2 + 36 (C)43(32)

MANHATTAN 135

GMAT

Chapter 3 Exponents & Roots

4 -8 (B) 22(7) (C)27(15)
35. 24 + 4 2

(A)7

Drill 6

Simplify the following expressions. All final answers should be integers.

36. 7 3 x ^ 2 7
37. a/2Xa/18

3 8 .#
a/3

39. a/5Xa/45
40, V5,000

V50

41. a/36Xa/4
a/128

42.
V2

a/54Xa/3
43.

&
a/640
44.
a/2Xa/5

a/30xa/12
45.

a/H)

Drill 7

Simplify the following roots. Not every answer will be an integer.

46. a/32
47. a/24
48. a/180
49. a/490
50. a/450
51. a/135
52. a/224
53. a/343

136 M AN HA TTA N

GMAT

Exponents & Roots Chapter 3

54. V208
55. V 432

Drill 8

Simplify the following roots. You will be able to completely eliminate the root in every question. Ex­
press answers as integers.

5 6 . n/ 3 6 2 + 1 5 2

5 7 . -\/352 - 2 1 2
58. V 6(56+ 5 7)

59. V84 + 8 5

60. V215+ 213- 212

61. V503- 5 0 2

1 1 - 1 12

62.
30

63. V57- 5 s + 5 4

NANHATTAN 137
GMAT

Exponents & Roots Chapter 3

Drill Sets Solutions

Drill 1

Simplify the following expressions by combining like terms. If the base is a number, leave the answer in
exponential form (i.e. 23, not 8).
I. x5 x x3= x (5+3) = x8
2 76 x 79= 7<6+9) = 7 15

3. - r = 5(5-3) = 52
5

4. U3)2= ^(3x2) = ^
5. 4-2 x 45= 4(_2+5) = 43

(-3)* o\(a-2)

7. (32)-3 = 3(2x' 3) = 3~6

114
8. _ = n«-*)

11*
9. X 2 X X 3 x X 5 = X (2 +3+5) = X 10
10. (52)*=5<2**> = 52*

Drill 2

Simplify the following expressions by combining like terms. If the base is a number, leave the answer in
exponential form (i.e. 23, not 8).

II. 34 x 32 x 3 = 3(4+2+1) = 37

12.——7 — = X<5+6- 2) = X9

X

56x 5 4* ^ (6 + 4 x —4) ^4x+2
13.

14. J/7 X J/8 X y 6= y 7 + 8 +("6))= y 9

MANHATTAN 139

GMAT

Chapter 3 Exponents & Roots

15< * *(4-(-3»= x 7
X

16. Z5Xigz—3 —£_(5+(-3)-(-8)) _—_10
£

17 o2x v 06*
—a(2*+6*-(-3j0) _ o8x+3^
1A 3-3,

18. (*2)6 x * 3 = * Cx6+3) = * (12+3) = * 1*

19. (zG)x x z3x= z(6xx+M = z{6x+3x) = z9x

5 x(5 Y _ ,-(3+(4xji)-(j/x3)) _ r(3+4j/-3j0 _ cy+5
(5,)3

Drill 3

Follow the directions for each question.

ii
21. 273 + 9 2 +-^r = V 2 7 + V 9 + - = 3 + 3 + 3 = 9

9° 1

22. We are looking for the answer with the lowest value, so we can focus only on answers that are nega­
tive as these answers have lower values than any positive answers.

(A) (-3)4will result in a positive number because 4 is an even power.

(B) - 3 3= -(3 3) = -2 7

The exponent is done before multiplication (by -1) because of the order of operations.

(C) (-3)-3 = — = =
(-3) -27 27

(D) (-2)3= - 8

(E) 2~6 = — The value of this expression is positive.

- 2 7 has the lowest value of the three answer choices that result in negative numbers. The correct answer
is (B).

23. 6~3 - —

The first two terms in the expression are in fact the same. Because these terms are equal, when the sec­
ond is subtracted from the first they cancel out leaving only the third term.

140 M ANHATTAN

GMAT

Exponents & Roots Chapter 3

I
24. " 2 n
, 5 , I ~ 5‘3III I f-1
I toUJ I UJ
I
|I
I

The correct answer is (E).

Note: when a problem asks you to find a different or more simplified version of the same thing, check
your work against the answer choices frequently to ensure that you don’t simplify or manipulate too far!

25. We are looking for values less than 1 so any expressions with negative values, zero itself, or values
between 0 and 1 will work.

l ll
( A ) - 0r = 3oO° x 2 2 1x4 4
Dividing a smaller positive number by a larger positive number will result in a number less than 1.

2-2 16
9

Dividing a larger positive number by a smaller positive number will result in a number greater than 1.

(C) ( -3 )3 - 2 7 27
(-5)2 25 25

This answer is negative; therefore, it is less than 1.

/2 V 2 2~2 32 9
(D) 3 2 4
Dividing a larger positive number by a smaller positive number will result in a number greater than 1.

(E) (-4)3= -6 4

This answer is negative; therefore, it is less than 1.

Drill 4

Simplify the following expressions by finding common bases.

26. 83 x 26= (23)3 x 26= 29 x 26= 215

27. 492 x 77—(72)2 x 77= 74 x 77 = 711

28 . 254 x 1253= (52)4 x (53)3= 58 x 59= 517

29. 9‘2 x 272= (32)-2 x (33)2= 3^ x 36= 32

30. 2~7 x 82 = 2~7 x (23)2 = 2~7 x 26= 2 '1

MANHATTAN 141

GMAT

Chapter 3 Exponents & Roots

Drill 5

Simplify the following expressions by pulling out as many common factors as possible.

31. Begin by breaking 6 down into its prime factors.

63+ 33=
(2 x 3)3+ 33=

( V ) & ) + 3s

Now each term contains (33). Factor it out.

(23)(33) + 33=
33(23+ 1) =
33(9) =
33(32) = 35

We have a match. The answer is A.

32. Both bases are powers of 3. Rewrite the bases and combine.

813+ 274 =
(34)3+ (33)4 =
312 + 312=
312(1 + 1) =
312(2)

We have a match. The answer is B.

33. Begin by breaking 15 down into its prime factors.

152- 52=
(3 x 5)2- 52=
(32)(52) - 52

Now both terms contain 52. Factor it out.

(32)(52) - 52=
52(32- 1) =
52(9 - 1) =
52(8)

We still don’t have a match, but we can break 8 down into its prime factors.

52(8) =

142 MANHATTAN

GMAT

Exponents & Roots Chapter 3

52(23)

We have a match. The answer is B.

34. Factor 43 out of the first four terms and factor 32 out of the last three terms.
43 + 43 + 43 + 43 + 32 + 32 + 32 =
43(1 + 1 + 1 + 1) + 32(1 + 1 + 1) =
43(4) + 32(3) =
44+ 33

We have a match. The answer is A.

35. Every base in the fraction is a power of 2. Begin by rewriting every base.

48 - 8 4 (22)8 - ( 2 3)4 216 - 2 12
24 + 4 2 “ 24 + (2 2)2 _ 24 + 2 4

The terms in the numerator both contain 212, and the terms in the denominator both contain 24. Factor
the numerator and denominator.

216 - 2 12 _ 212(24 -1 ) _ 212(16 -1 ) _ 212(15)
24 + 2 4 ~ 24(1 + 1) ” 24(2) ~ 25

At this point, we can combine the 2’s in the numerator and the denominator.

2

We have a match. The answer is C.

Drill 6

Simplify the following expressions. All final answers should be integers.

36. V 3 x V 2 7 = V 3 ^ 2 7 = V 8 l = 9

37. V 2 x ^ = %/2x 1 8 = ^ = 6

38. V48 [48 = y/l6 = 4
r ■= J —
S V3

39. V 5 x V 4 5 = V 5 x 4 5 = V 2 2 5 = 1 5
, A 000 15,000 I-----
40. v ’ = ./ - ----- = V l 0 0 = 1 0

>^0 V 50

MANHATTAN 143

GMAT

Chapter 3 Exponents & Roots

41. V 3 6 x V 4 = V 3 6 x 4 = V l 4 4 = 1 2 OR

^ 3 6 x 7 4 = 6 x 2 = 12
42 m

V2 V 2

44. = J 6 « [ = J M = V 64=8

^ 2 x^5 V2x5 V 10

V Bp® |30HT2= v 5 = 6

a /Io V io

Drill 7

Simplify the following roots. Not every answer will be an integer.

46. a/32 = V 2 x 2 x 2 x 2 x 2 = a/2x 2 x V 2 x 2 X a/2 = 2 x 2 X a/2 = 4 a/2
47. a/24 = a/2x 2 x 2 x 3 = a/2x 2 X a/2><3=2a/6
48. Vi 8 0 = a/2x 2 x 3 x 3 x 5 = a/2x 2 x V3x 3 x >/ 5 = 2 x 3 x >/5 = 6 a/5
49. a/490 = a/2x 5 x 7 x 7 = a/7x 7 x 7 2 x 5 = 7 a/10
50. a/450 = a/2x 3 x 3 x 5 x 5 = a/3x 3 * a/5x 5 x V 2 = 3 x 5 x a/2 = 1 5 a/2
51. a/135 = a/3x 3 x 3 x 5 = a/3x 3 x V3x ? = 3a/15
52. V224 = a/2x 2 x 2 x 2 x 2 x 7 = ^ 2 x 2 x V2x 2 x -v/2x 7 = 2 x 2 x 714 = 4 a/i 4
53. a/343 = a/7x 7 x 7 = a/7x 7 xa/7 = 7 V 7
54. a/208 = a/2x 2 x 2 x 2 x 1 3 = a/2x 2 X a/2x 2 X a/ 1 3 = 2 x 2 x Vi 3 = 4 a/13
55. a/432 = a/2x 2 x 2 x 2 x 3 x 3 x 3 = V 2 x 2 X a/2x 2 x V3x 3 x V 3 = 2 x 2 x 3 x ^ = 12a/3

Drill 8

Simplify the following roots. You will be able to completely eliminate the root in every question. Ex­
press answers as integers.

56. Pull out the greatest common factor of 362and 152, namely 32, to give
yj32(l2 2 + 5 2) = >/32(l44 + 25) = \J32(169) . Both 32 and 169 are perfect squares (169 = 132), so
a/32(l 69) = a/32(l 32) = 3 x 13 = 39 .

57. Pull out the greatest common factor of 352 and 212, namely 72, to give
>/72(52 - 3 2) = a/7 2(25 —9) = >/72(16). Both 72 and 16 are perfect squares (16 = 42), so
a/72(16)=a/72(42) = 7 x 4 = 28.

144 MANHATTAN

GMAT

Exponents & Roots

58. Pull out the greatest common factor of 56 and 57, namely 56, to give
V6(56(1 + 5)) = >/6(56(6)) = V62(56) . Both 62 and 56are perfect squares (56= 53x 53), so
^/62(5 6 ) = 6 x 5 3 = 6 x 1 2 5 = 7 5 0 .

59. Pull out the greatest common factor of 84 and 85, namely 84, to give yjs4(l + 8) = yfs\ 9) = V84(32) .
Both 84 and 32 are perfect squares (84= 82 x 82), so a/84(32) = 82 x 3 = 64 x 3 = 192.

60. Pull out the greatest common factor of 215, 213, and 212, namely 212, to give
V212(23 + 2 - l ) = -v/212(8 + 2 - 1 ) = V2,2(9) = sj2l2&2) . Both 212and 32 are perfect squares (212= 26 x
26), so V2I2(32) = 2 6 x3 = 6 4 x 3 = 192.

61. Pull out the greatest common factor of 503and 502, namely 502, to give
V502(50 -1 ) = a/502(49) = V502(72). Both 502and 72are perfect squares, so ^502(72) = 5 0 x 7 = 350.
62. First focus on the numerator of the fraction under the radical and pull out the greatest common

factor of l l 4 and l l 2, namely l l 2, to give ^ ~^ ^ 2— . The denominator

(30) divides evenly into 120: = V ^ ( 4 ) = \/ll2(22) . Both l l 2and 22 are perfect squares, so
-y/ll2(22) = 11x2 = 22.

63. Pull out the greatest common factor of 57, 55, and 54, namely 54, to give
V54(53 - 5 + 1) = V54(1 2 5 -5 + l) = ^54(l2 l) = a/54(112) . Both 54 and l l 2 are perfect squares (54= 52 x
52), so -y/54(112) = 5 2 x l 1 = 25x11 = 275 .

MANHATTAN

GMAT

Foundations of GMAT Math

Fractions

B a s ic s o f F ra c tio n s

Add Fractions with the Same Denominator: Add the Numerators
Add Fractions with Different Denominators:
Find a Common Denominator First

Compare Fractions: Find a Common Denominator (or Cross-Multiply)
Change an Improper Fraction To a Mixed Number: Actually Divide
Change a Mixed Number To an Improper Fraction: Actually Add
Simplify a Fraction: Cancel Common Factors on Top and Bottom
Multiply Fractions: Multiply Tops and Multiply Bottoms
(But Cancel First)
Square a Proper Fraction: It Gets Smaller
Take a Reciprocal: Flip the Fraction
Divide by a Fraction: Multiply by the Reciprocal
Addition in the Numerator: Pull Out a Common Factor
Addition in the Numerator: Split into Two Fractions (Maybe)
Addition in the Denominator: Pull Out a Common Factor
But Never Split
Add, Subtract, Multiply, Divide Nasty Fractions: Put Parentheses On
Fractions Within Fractions: Work Your Way Out

m p t e r 4 .-

Fractions

In This Chapter:

• Rules for m anipulating fractions

Basics of Fractions

To review, a fraction expresses division.

The numerator on top is divided by the denominator on bottom.

N um erator s .

Fraction line -----► 3 — 3 ■=■4
D enom inator
S —
4
'

T h ree fou rths is th ree divided b y four.

The result o f the division is a num ber. If you punch “3 4 = ” into a calculator, you get the decimal
0 .7 5 . But you can also th in k o f 0 .7 5 as —3 , because —3 and 0.7 5 are two different names for the same

44
num ber. (W e’ll deal w ith decimals in the next chapter.)

Fractions express a part-to-whole relationship.

3 p a r t ^ ----- 3 pieces = part
4 pieces = whole
4 whole

Chapter 4 Fractions

In this picture, a circle represents a whole unit— a full pizza. Each pizza has been divided into 4 equal
parts, or fourths, because the denominator of the fraction is 4. In any fraction, the denominator tells
you how many equal slices a pizza has been broken into. In other words, the denominator tells you the
size of a slice.

The numerator of the fraction is 3. This means that we actually have 3 slices of the pizza. In any frac­
tion, the numerator tells you how many slices you have.

Together, we have three fourths: three slices that are each a fourth of a whole pizza.

Since fractions express division, all the arithmetic rules of division apply. For instance, a negative di­
vided by a positive gives you a negative, and so on.

-3 + 4 = -0.75 -^7 = 3-s-( -4 ) = -0.75

So —- 3 and —3 represent the same number. We can even write that number as ——3 . Just don’t mix up

the negative sign with the fraction bar.
PEMDAS also applies. The fraction bar means that you always divide the entire numerator by the
entire denominator.

^ r t l = (3x2 + y ) ^ ( 2 / - z )

The entire quantity 3x2 + y is being divided by the entire quantity 2y 2 —z .

If you rewrite a fraction, be ready to put parentheses around the numerator or denominator to preserve
the correct order of operations.

Finally, remember that you can’t divide by 0. So a denominator can never equal zero. If you have a
variable expression in the denominator, that expression cannot equal zero. If a problem contains the

3<%’2 -j- y
fraction — -— —, then we know that -—z cannot equal zero. In other words, we know

2y —z

ly1- #^ 0 or 2y2^ z

If the GMAT tells you that something does not equal something else (using the ^ sign), the purpose is
often to rule out dividing by 0 somewhere in the problem.

To compare fractions with the same denominator, compare the numerators. The numerator tells you
how many pieces you have. The larger the numerator, the larger the fraction (assuming positive
numbers everywhere). You have more slices of pie.

150 MANHATTAN
GMAT

Fractions Chapter4

To compare fractions w ith the same numerator, compare the denominators. Again assume positive
numbers everywhere. The larger the denominator, the sm aller the fraction. Each slice o f pie is
smaller. So the same num ber o f smaller slices is smaller.

1>
7

If the num erator and denom inator are the same, then the fraction equals 1.

4 equals 4-7-4 1
4
four divided which one
Four by four, equals
fourths

If the num erator is larger than the denom inator (again, assume positive numbers), then you have more
than one pizza.

5 equals 5+4 I+
4
five divided which one plus
Five by four, equals
fourths

A nother way to write 1 -t— is 1— (“one and one fourth”). This is the only time in G M A T m ath when
44

we put two things next to each other (1 and —) in order to add them . In all other circumstances, we

4

mean multiplication when we put two things next to each other.

M ANHATTAN 151

GMAT

Fractions

A mixed number such as 1— contains both an integer part (1) and a fractional part —. You can always
44

rewrite a mixed num ber as a sum of the integer part and the fractional part.

33
3 —= 3 + —

In an improper fraction such as —, the numerator is larger than the denominator. Im proper fractions
4

and mixed num bers express the same thing. Later we’ll discuss how to convert between them .

A proper fraction such as — has a value between 0 and 1. In a proper fraction, the num erator is
4

smaller than the denominator.

Add Fractions with the Same Denominator: Addthe Numerators

The num erator o f a fraction tells you how many slices o f the pizza you have. So when you add fractions,
you add the num erators. You just have to make sure that the slices are the same size— in other words,
th at the denom inators are equal. Otherwise you aren’t adding apples to apples.

+3 4
5 5

In words, one fifth plus three fifths equals four fifths. The “fifth” is the size o f the slice, so the denom i­
nator (5) doesn’t change.
Since 4 = 1 + 3 , you can write the fraction with 1 + 3 in the numerator.

1 3 1+3 4
5 + 5 _ 5 ~~ 5
The same process applies w ith subtraction. Subtract the numerators and leave the denom inator the
same.
9 4 _ 9-4 _ 5
14 141414

M ANHATTAN

GMAT

Fractions Chapter 4

If variables are involved, add or subtract the same way. Just make sure that the denominators in the
original fractions are equal. It doesn’t matter how complicated they are.

3a x 4a _ 3 a + 4a _ 7 a 5x 2xl 5x2 —2x2 3x2
T + T ~ b ~~b z+w z+w z+w z+w

If you can’t simplify the numerator, leave it as a sum or a difference. Remember that the denominator
stays the same: it just tells you the size of the slices you’re adding or subtracting.

3n 5m 3n —5m

2w 2w 2w5

If you... Then you... Like this:

2 3 2+3

Add or subtract fractions Add or subtract the 77 7
that have the same numerators, leaving the
denominator _5
denominator alone

7

Check Your Skills
3x 7x

+-y--z-- 2 -y--z-- 2

Answers can befound on page 179.

Add Fractions with Different Denominators: Find a Common
Denominator First

Consider this example:

I 1=

4+8

The denominators (the sizes of the slices) aren’t the same, so you can’t just add the numerators and call
it a day.

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To add these fractions correctly, we need to re-express one or both o f the fractions so that the slices are

the same size. In other words, we need the fractions to have a common denominator— that is, the same

denominator. Once they have the same denominator, we can add the numerators and be finished.

Since a fourth o f a pizza is twice as big as an eighth, we can take the fourth in the first circle and cut it
in two.

12

one fourth two eighths

W e have the same am ount o f pizza— the shaded area hasn’t changed in size. So one fourth ( \_y must

v4,

equal two eighths

W h en we cut the fourth in two, we have twice as many slices. So the num erator is doubled. But we’re
breaking the whole circle into twice as many pieces, so the denom inator is doubled as well. If you
double both the num erator and the denominator, the fraction’s value stays the same.

1 1x2
4 4x2

W ithout changing the value o f —1 , we have renamed it —2 . Now we can add 3 it to—
48 8
5

+

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Fractions Chapter 4

All in one line:

I 3_lx2_ 3 _ 2 3 _5
4 8 ~~ 4 x 2 8 ~ 8 8 ~ 8

To add fractions with different denominators, find a common denominator first. That is, rename
the fractions so that they have the same denominator. Then add the new numerators. The same holds
true for subtraction.

How do you rename a fraction without changing its value? Multiply the top and bottom by the same
number.

1 1x2 8 3 _ 3 x 2 5 _ 75 5 x 7 35
4 4x2 4 ~ 4 x 2 5 ~ 100 12 1 2 x 7 84

Heres why this works. Doubling the top and doubling the bottom of a fraction is the same as multiply-
ing the fraction by —2 . (More on fraction multiplication later.)

2
Moreover, — is equal to 1. And multiplying a number by 1 leaves the number the same. So if we

multiply by —2 , we aren’t really changing the number. We’re just changing its look.

I - I 2 - 1x2 _ 2
4 ~ 4 2 ~~ 4 x 2 ~~ 8

You can rename fractions that have variables in them, too. You can even multiply top and bottom by
the same variable.

* _ x x 2 _ xX2 2x a _ a b _ a x b _ ab
y y 2 yX 2 2y 2 ~ 2 ~b~ 2 x b ~ 2b

Just make sure the expression on the bottom can never equal zero, of course.

If you... Then you... Like this:

Want to give a fraction a Multiply top and bottom of 1 _ 1x2 _ 2
different denominator but the fraction by the 4 4x2 8
keep the value the same same number

Say you have this problem:

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Fractions

What should the common denominator of these fractions be? It should be both a multiple of 4 and a
multiple of 3. That is, it should be a multiple that 3 and 4 have in common. The easiest multiple to pick
is usually the least common multiple (LCM) of 3 and 4. Least common multiples were discussed on
page 77.

The least common multiple of 4 and 3 is 12. So we should rename our two fractions so that they each
have a denominator of 12.

1 _ 1x3 _ 3 1 _ 1x4 _ 4
4 _ 4 x 3 _ 12 3 ~ 3 x 4 12

Once you have a common denominator, add the numerators:

1 , 1 _ 1x3 | 1x4 _ 3 | 4 _ 7
4 3 ~~ 4 x 3 3 x 4 ~ 12 12 ~ 12

The process works the same if we subtract fractions or even have more than two fractions. Try this
example:

5 2_3 =
6+9 4

First, find the common denominator by finding the least common multiple. All three denominators (6,
9, and 4) are composed of 2’s and 3’s.

6 = 2x3 9= 3x3 4= 2x2

The LCM will contain two 2’s (because there are two 2’s in 4) and two 3’s (because there are two 3’s in
9).

2 x 2 x 3 x 3 = 36

64 9
To make each of the denominators equal 36, multiply the fractions by —, —, and —, respectively.

5 _ 5 x 6 30 2 2x4 _ 8 3 _ 3 x 9 _ 27
6 ~ 6 x 6 _ 36 9 ~ 9 x 4 36 4 _ 4 x 9 _ 36

Now that the denominators are all the same, we can add and subtract normally.

5 2 _ 3 _ 30 8 27 _ 30 + 8 - 2 7 _ 11
6 9 4 ~ 36 36 36 ~ 36 ~ 36

The process works even if we have variables. Try adding these two fractions:

MANHATTAN

GMAT

Fractions

First, find the common denominator by finding the least common multiple of x and 2x. The LCM is
2xysince you can just multiply the x by 2. So give the first fraction a denominator of 2x, then add:

2 t 3 _ 2x2 ( 3 _ 4 ^ 3 _ 4+ 3 _ 7
x 2x x x 2 2x 2x 2x 2x 2x

If you... Then you... Like this:

Add or subtract fractions Put the fractions in terms 1 1 1x5 1x3
with different denominators of a common denominator, 3 5 3x5 5x3

then add or subtract _5_ 3__8_

15 + 15 15

Check Your Skills

2 .1 +i =
24

3. — =
38

Answers can befound on page 179.

Compare Fractions: Find a Common Denominator (or Cross-Multiply)

Earlier in the chapter, we talked a little about comparing fractions. If two fractions have the same de­
nominator, then you compare the numerators.

Now you can compare any two fractions. Just give them the same denominator first.

Which is larger, —4 or —3 ?
75

First, find a common denominator and re-express the fractions in terms of that denominator. The least
common multiple of 7 and 5 is 35, so convert the fractions appropriately:

4 _ 4 x5 _ 2° 3 _ 3 x 7 _ 21
7 _ 7 x 5 ~ 35 5 "5 x 7 "35

3
Now you can tell at a glance which fraction is larger. Since 21 is greater than 20, you know that — is

greater thuan —4 . 5

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ir 4 Fractions

A good shortcut is to cross-multiply the fractions. Heres how:

(1) Set them up near each other. 34
(2) Multiply “up” the arrows as shown.
57
Be sure to put the results at the top.
7 x 3 = 21 3 4 5 x 4 = 20

TT

(3) Now compare the numbers you get. The side with the bigger number is bigger.

21 > 20, so —3 is greater than —4 .
57

This process generates the numerators we saw before (21 and 20). You don’t really care about the com­
mon denominator itself (35) so cross-multiplying can be fast.

If you... Then you... Like this:
Want to compare fractions
Put them in terms of a (2 ^ 4 3 ^ 27
common denominator,
or just cross-multiply 9X 7

Check Your Skills

For each of the following pairs of fractions, decide which fraction is greater.

41

Answers can befound on page 179.

Change an Improper Fraction Toa Mixed Number: Actually Divide

What is —13 as a mixed number? Note that —13 is an improper fraction, because 13 > 4.
44

Since the fraction bar represents division, go ahead and divide 13 by 4. Try doing this by long division:

3
4)13

12
1

4 goes into 13 three times, with 1 left over. 3 is the quotient, representing how many whole times the
denominator (4) goes into the numerator (13).

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Fractions Chapter 4

Meanwhile, 1 is the remainder.

13

So — equals 3, plus a remainder of 1. This remainder of 1 is literally “left over” the 4, so we can write
the whole thing out:

—=3+—
44

As a mixed number, then, —13 equals 3 —1 .
44

To convert an improper fraction to a mixed number, actually divide the numerator by the denom­
inator. The quotient becomes the integer part of the mixed number. The remainder over the denomina­
tor becomes the left-over fractional part of the mixed number.

To do the division, look for the largest multiple of the denominator that is less than or equal to the
numerator. In the case of —13 , we should see that 12 is the largest multiple of 4 less than 13. 12 = 4 x 3 ,

4
so 3 is the quotient. 13 - 12 = 1, so 1 is the remainder.

Heres another way to understand this process. Fraction addition can go forward and in reverse.

r orward,: —2 + —4 = -2--+---4-= —6 Rneverse: —6 = --2--+--4---= 2— I—4
77 7 7 7777

In other words, you can rewrite a numerator as a sum, then split the fraction. Try this with —13 .
4

Rewrite 13 as 12 + 1, then split the fraction:

13 _ 12 + 1 _ 12 J.
4 ~ 4 “ 4+4

Since —12 = ^12 + 4/o= 3,wei_have —13 = -1--2- h—= 3 + —= 3 —1.11
4 44 444

You still need to find the largest multiple of 4 that’s less than 13, so that you can write 13 as 12 + 1.

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Chapter 4 Fractions

If you... Then you... Like this:

Want to convert an Actually divide the 13=13 + 4
improper fraction to a numerator by the 4

mixed number denominator = 3 rem ainder 1

OR 13 _ 12 + 1
Rewrite the numerator as a 44
sum, then split the fraction
_ 12 1
“T +4

=4

Check Your Skills

Change the following improper fractions to mixed numbers.

11
6- 7

7 12® Answers can befound on page 179.

11

Change a Mixed NumberToan Improper Fraction: Actually Add

2
What is 5— as an improper fraction?

22
First, rewrite the mixed number as a sum. 5—= 5 + —.

33
Now, let’s actually add these two numbers together by rewriting 5 as a fraction. You can always write
any integer as a fraction by just putting it over 1:

5 =— This is true because 5 ^ 1 = 5.
1

22 5 2

So 5 —= 5 + —= —+ —.At this point, you’re adding fractions with different denominators, so find a
3 3 13

common denominator.

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