1. Section 9.2
Rational Exponents

2. Objectives Simplify expressions of the form a1/n
Simplify expressions of the form am/n
Convert between radicals and rational exponents
Simplify expressions with negative rational exponents
Use rules for exponents to simplify expressions.
Simplify radical expressions

3. Objective 1: Simplify Expressions of the Form a1/n
The Definition of x1/n : A rational exponent of indicates the nth root of its base.
If n represents a positive integer greater than 1 and represents a real number,
We can use this definition to simplify exponential expressions that have rational exponents with a numerator of 1. For example, to simplify , we write it as an equivalent expression in radical form and proceed as follows:
Read as “x to the 1/n power equals the nth root of x.”
Thus, = 2.

4. Objective 1: Simplify Expressions of the Form a1/n
Summary of the definitions of : If n is a natural number greater than 1 and x is a real number,

5. EXAMPLE 1
Evaluate :
Strategy First, we will identify the base and the exponent of the exponential expression. Then we will write the expression in an equivalent radical form using the rule for rational exponents
Why We can then use the methods from Section 7.1 to evaluate the resulting square root, cube root, fourth root, and fifth root.

6. EXAMPLE 1
Evaluate :
Solution

7. Objective 2: Simplify Expressions of the Form am/n
The Definition of xm/n : If m and n represent positive integers, (n ≠ 1) and represents a real number,

8. Objective 2: Simplify Expressions of the Form am/n
Because of the previous definition, we can interpret xm/n in two ways:
xm/n means the nth root of the mth power of x.
xm/n means the mth power of the nth root of x.
We can use this definition to evaluate exponential expressions that have rational exponents with a numerator that is not 1. To avoid large numbers, we usually find the root of the base first and then calculate the power using the rule

9. Evaluate:
EXAMPLE 3
Strategy First, we will identify the base and the exponent of the exponential expression. Then we will write the expression in an equivalent radical form using the rule for rational exponents
Why We know how to evaluate square roots, cube roots, fourth roots, and fifth roots.

10. EXAMPLE 3 Solution Evaluate:
a. To evaluate 322/5, we write it in an equivalent radical form. The denominator of the rational exponent is the same as the index of the corresponding radical. The numerator of the rational exponent indicates the power to which the radical base is raised.

11. Evaluate:
EXAMPLE 3
Solution

12. Evaluate:
EXAMPLE 3
Solution

13. Objective 3: Convert Between Radicals and Rational Exponents
We can use the rules for rational exponents to convert expressions from radical form to exponential form, and vice versa.

14. EXAMPLE 5
Write as an exponential expression with a rational exponent.
Strategy We will use the first rule for rational exponents in reverse:
Why We are given a radical expression and we want to write an equivalent exponential expression.

15. EXAMPLE 5
Write as an exponential expression with a rational exponent.
Solution
The radicand is 5xyz, so the base of the exponential expression is 5xyz. The index of the radical is an understood 2, so the denominator of the fractional exponent is 2.

16. Objective 4: Simplify Expressions with Negative Rational Exponents
Definition of x –m/n: If m and n are positive integers, is in simplified form, and x1/n is a real number, then

17. EXAMPLE 7 Strategy We will use one of the rules
Simplify. Assume that x can represent any nonzero real number.
EXAMPLE 7
Strategy We will use one of the rules
to write the reciprocal of each exponential expression and change the exponent’s sign to positive.
Why If we can produce an equivalent expression having a positive rational exponent, we can use the methods of this section to simplify it.

18. Simplify. Assume that x can represent any nonzero real number.
EXAMPLE 7
Solution

19. Objective 5: Use the Rules for Exponents to Simplify Expressions
We can use the rules for exponents to simplify many expressions with fractional exponents. If all variables represent positive real numbers, absolute value symbols are not needed.

20. EXAMPLE 8
Simplify. All variables represent positive real numbers. Write all answers using positive exponents only.
Strategy We will use the product, power, and quotient rules for exponents to simplify each expression.
Why The familiar rules for exponents discussed in Chapter 5 are valid for rational exponents.

21. EXAMPLE 8
Simplify. All variables represent positive real numbers. Write all answers using positive exponents only.
Solution

22. EXAMPLE 8
Simplify. All variables represent positive real numbers. Write all answers using positive exponents only.
Solution

23. Objective 6: Simplify Radical Expressions
We can simplify many radical expressions by using the following steps.
Change the radical expression into an exponential expression.
Simplify the rational exponents.
Change the exponential expression back into a radical.

24. EXAMPLE 10
Simplify:
Strategy We will write each radical expression as an equivalent exponential expression and use rules for exponents to simplify it. Then we will change that result back into a radical.
Why When the given expression is written in an equivalent exponential form, we can use rules for exponents and our arithmetic skills with fractions to simplify the exponents.

25. EXAMPLE 10
Simplify:
Solution a. b.

26. EXAMPLE 10
Simplify:
Solution

27. EXAMPLE 10
Simplify:
Solution