Section 7.2

• So far, we have only worked with integer exponents.

• In this section, we extend exponents to rational numbers as a shorthand notation when using radicals.

• The same rules for working with exponents will still apply.

Recall that a cube root is defined so that

abba 33 ifonly

However, if we let b = a1/3, then aaaab 133/133/13 )(

Since both values of b give us the same a,

33/1 aa

nn aa /1

n aIf n is a positive integer greater than 1 and is a real number, then

Use radical notation to write the following. Simplify if possible.

3381 4 44 4/181

5/11032x 25 1055 10 2232 xxx

3/1716x 3233 633 743 7 2222216 xxxxxx

Example

We can expand our use of rational exponents to include fractions of the type m/n, where m and n are both integers, n is positive, and a is a positive number,

mnn mnm aaa /

Use radical notation to write the following. Simplify if possible.

3/48 4

3 8

3/773x 3 773x

Example

32 73)73( xx

4

3 32 42 16

33 6 7373 xx

Now to complete our definitions, we want to include negative rational exponents.

If a-m/n is a nonzero real number,

nmnm

aa

// 1

Use radical notation to write the following. Simplify if possible.

3/264 3/264

1

4/516 4/516

1

Example

23 64

1

23 34

1

24

116

1

54 42

1

52

1

32

1

All the properties that we have previously derived for integer exponents hold for rational number exponents, as well.

We can use these properties to simplify expressions with rational exponents.

Use properties of exponents to simplify the following. Write results with only positive exponents.

33/25/132 x 25/332 x

3/2

2/14/1

a

aa 3/22/14/1a

Example

23

5 52 x 232 x 28x

12/812/612/3a 12/11a 12/11

1

a

Use rational exponents to write as a single radical.

Example

253 2/13/1 25 6/36/2 25 6/132 25 6 200