# Rational Exponents

Definition of

If a represents a real number and n≥2 is an integer, then

If n is odd and

• a is positive, then
is positive.

• a is negative, then
is negative.

• a is zero, then
is zero.

If n is even and

• a is positive, then
is positive.

• a is negative, then
is not a real number

• a is zero, then
is also zero.

**Example 1: **Use radical notation to rewrite each
expression.

Simplify, if possible.

Example 2: Rewrite each expression using rational exponents.

**Definition of **

If represents a real
number and is a positive rational number,

n≥2, then

Note that if n is even and a is negative,does
not represent a real

number and is not a real number.

Example 3: Use radical notation to rewrite each of the
following

and then simplify.

Example 4: Rewrite with rational exponents.

**Definition of **

If is a nonzero real number, then

Example 5: Rewrite each of the following with a positive

exponent. Simplify, if possible. Assume all variables represent

nonnegative quantities.

**Properties of Rational Exponents**

If m and n are rational exponents, and a and b are real
numbers for

which the following expressions are defined, then

Example 6: Simplify the following expressions with
rational

exponents. Express all answers with positive exponents.

Assume all variables represent nonnegative quantities.

**Simplifying Radical Expressions Using Rational
Exponents**

To simplify a radical expression by using rational
exponents:

1. Rewrite each radical expression as an exponential

expression with a rational exponent.

2. Simplify using properties of rational exponents.

3. Rewrite your answer in radical notation when rational

exponents still appear.

Example 7: Use rational exponents to simplify. Assume all

variables represent nonnegative quantities.

**Application of Rational Exponents**

Example 8: The function
models the number of

calories per day, f(x), that a person needs to maintain life in

terms of that person’s weight, x, in kilograms. (1 kilogram is

approximately 2.2 pounds.) Use the model and a calculator to

find how many calories per day are required to maintain life for

a person who weighs 55 kilograms (about 121 pounds). Round

your answer to the nearest calorie.

Example 9: Use your calculator to evaluate the following
to

three decimal places.

**Answers Section 10.2**

Example 1:

Example 2:

Example 3:

a. 64

b. 4

c. Not a real number

d. −8

Example 4:

Example 5:

Example 6:

Example 7:

Example 8:

a. x = 55 kg., f(55) 1414 calories

Example 9:

a. 3.911

b. 75.421

c. 20.983