7.2 Properties of Rational Exponents

3/4/2013

Example 1 Use Properties of Rational Exponents

a. 62/3 • 61/3 = 6(2/3 + 1/3) = 63/3 = 61 = 6

b. (33/4)4 = 3(3/4 4)• = 33 = 27

c. (16 • 25)1/2 = 20= 161/2 • 251/2 = 4 • 5

d. 8 1/3–1

=81/3

=2

1

e.71/2

75/2

= 7(5/2 1/2)– = 74/2 = 72 = 49

Fourth RootPerfect Fourth

1 = 14

16 = 24

81 = 34

256 = 44

625 = 54

4√1=14√16=24√81=34√256=4

4√625=5

Fifth RootPerfect Fifth1 = 15

32 = 25

243 = 35

1024 = 45

3125 = 55

5√1=15√32=25√243=35√1024=45√3125=5

Simplifying

2552 33 3 82

44 4 6255 55 5 2433

=5=2=5=3

In general aan n

Properties of Radicals

Product property:

Quotient property:

Example 2 Use Properties of Radicals

a. 33 • 93

= 3 Simplify.

= 4

3

48 Quotient property of radicals

= 2 Simplify.

= 33 • 9 Product property of radicals

= 33 • 3 Factor.• 3

b.4 48

34

= 24 • 2 Divide and factor.• 2 • 2

4 16

3 33

4 42

𝑜𝑟 3√3 ∙ 9= 3√27=3

2

Example 3 Write Radicals in Simplest Form

a. 3 40 = 83 • 5 Factor out a perfect cube.

= Separate the product83 • 53

= 2 Simplify.53

5√64b. = Factor out a perfect fifth.

=

Checkpoint

Simplify the expression.

Use Properties of Radicals and Rational Exponents

ANSWER 2

ANSWER 2

ANSWER 3

ANSWER 3

1. 23 • 43

2. 45 • 85

3. 543

23

4. 813

33

Example 5 Simplify Expressions with Variables

a. 9x 6

Simplify the expression. Write your answer using positive exponents only. Assume all variables are positive.

= 3x 3 Simplify.

b. 4y 6 ( )1/2 = 41/2 y 6 ( )1/2 Power of a product property

= Power of a power propertyy (6 · 1/2)

= 2y 3 Simplify.

= Break into perfect square factors.

Example 5 Simplify Expressions with Variables

c. 3

y 6

x 3

=x

y 2 Simplify.

d.ac 2

3a 3/2 c

–

Quotient of powers property3a (3/2 1)c [1 ( 2)]= – – –

Simplify.3a 1/2c 3=

Factor out perfect cube factors.¿3√𝑥3

3√𝑦3 ∙ 𝑦3

¿3√𝑥3

3√𝑦3 ∙3√𝑦3

𝑒 .√𝑥 𝑦2𝑧 3 Break into perfect square factors.

Example 5

Take out perfect squares.

Checkpoint

Simplify the expression. Write your answer using positive exponents only. Assume all variables are positive.

ANSWER 5y 2

ANSWER 2uv 3

1. 25y 4

ANSWER 2x 2/3z 3

Simplify Expressions with Variables

2. y 2

x 6ANSWER

yx 3

3. 8u 3v 9( )1/3

4. 2xx 1/3z 3–

Homework: Prac A WS 7.2 #