Rational Exponents: Definition, Simplification, and Properties, Summaries of Algebra

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Rational Exponents

Section 4.

• How should we define , n is a counting number?
• Exponent property is true if m = ½ n = 2
• (–3)^2 = 9 and 3^2 = 9
• Suggests that 9 ½^ = 3
• The nonnegative number 3 is the principal second root , or principle square root, of 9, written
• If m = , n = 3:

Definition of Rational Exponents

Definition of Rational Exponents

Introduction

b^1 n ( ) m n mn b = b

(^13)

1 3 1 3 33 1 8 b b b

  =^ =^ =

For the counting number n , where n ≠ 1,

• If n is odd, then is the number whose n th power is b , and we call the nth root of b****.
• If n is even and , then is the nonnegative number whose n th power is b , and we call the principle square root of b****.
• If n is even and b < 0 , then is not a real number

may be represented by

Definition of Rational Exponents

Definition of Rational Exponents

Definition

b^1 n

b ≥ 0

b^1^ n b^1^ n b^1 n

b^1 n b^1^ n n^ b.

Simplify.

Simplifying Expressions Involving Rational Exponents

Definition of Rational Exponents

Example

( )

( )

1 2 1 3^ 1 3 1 4 1 4 1 4

Solution

( ) ( )

1 2 2 1 3 3 1 3 3

1. 25 5, since 5 25
   2. 64 4, since 4 64
2. 64 4, since 4 64

For the counting numbers m and n , where n ≠ 1 and b is any real number for which is a real number,

A power of the form or is said to have a rational exponent.

Definition: Rational Exponent

Definition of Rational Exponents

Definition

b^1 n

( ) ( ) 1 1

1

m n n m^ m n

m n n

b b b

b b b

−

bm n b −^ m n

Simplify.

Simplifying Expressions Involving Rational Exponents

Definition of Rational Exponents

Example

1. 253 2 2. ( − 27 ) 2 3^ 3. 32 −2 5 4. ( − 8 )−5 3

Solution

3 2 1 2 3 3

2 3 1 3 2 2

For

find the following:

Simplifying Expressions Involving Rational Exponents

Definition of Rational Exponents

Example

f (^) ( x (^) ) = 64 , x g (^) ( x (^) ) = 3 16( )^ x^ , and h x ( ) = −5 9( ) x ,

Solution

f ^ ^ g ^ ^ h ^ −       

If m and n are real rational numbers and b and c are any real number for which bm^ , bn^ and c n^ are real numbers

Simplifying Expressions Involving Rational Exponents

Definition of Rational Exponents

Solution Continued

Properties