Intermediate Algebra (11th edition)

(d) (e)

(f ) -

B

3

m^6

125

= -

23 m^6

23125

= -

m^2

5

B

5

x

32

=

25 x

2532

=

25 x

B^2

3

7

216

=

237

23216

=

237

6

SECTION 8.3 Simplifying Radical Expressions 445

NOW TRY

Think: 23 m^6 =m^6 >^3 =m^2

OBJECTIVE 3 Simplify radicals.We use the product and quotient rules to sim-

plify radicals. A radical is simplifiedif the following four conditions are met.

Conditions for a Simplified Radical

1. The radicand has no factor raised to a power greater than or equal to the index.

2. The radicand has no fractions.

3. No denominator contains a radical.

4. Exponents in the radicand and the index of the radical have greatest common

factor 1.

Simplifying Roots of Numbers

Simplify.

(a)

Check to see whether 24 is divisible by a perfect square (the square of a natural

number) such as 4, 9, 16,.... The greatest perfect square that divides into 24 is 4.

Factor; 4 is a perfect square.

Product rule

(b)

As shown on the left, the number 108 is divisible by the perfect square 36. If this

perfect square is not immediately clear, try factoring 108 into its prime factors, as

shown on the right.

Factor.

Product rule

Product rule

Multiply.

(c)

No perfect square (other than 1) divides into 10, so cannot be simplified

further.

210

210

= 623

= 2 # 3 # 23 222 =2, 232 = 3

= 623 236 = 6 = 222 # 232 # 23

= 236 # 23 = 222 # 32 # 3 a^3 =a^2 #a

= 236 # 3 = 222 # 33

2108 2108

2108

= 226 24 = 2

= 24 # 26

= 24 # 6

224

224

EXAMPLE 4

NOW TRY
EXERCISE 3
Simplify. Assume that all
variables represent positive
real numbers.

(a) (b)

(c) (d)

(e)-
B

5

m^15

243

B

4

t

B^16

3 -

27

1000

B

5

B^144

49

36

NOW TRY ANSWERS

3. (a) (b) (c)

(d) (e)-

m^3

3

24 t

2

• 3
  10

25

12

7

6