1. Copyright © Cengage Learning. All rights reserved. Roots, Radical Expressions, and Radical Equations 8

2. Copyright © Cengage Learning. All rights reserved. Section 8.7 Rational Exponents

3. 3 Objectives Simplify a numerical expression containing a rational exponent with a numerator of 1. Simplify a numerical expression containing a rational exponent with a numerator other than 1. Apply the rules of exponents to an expression containing rational exponents. 1 1 2 2 3 3

4. 4 Simplify a numerical expression containing a rational exponent with a numerator of 1 1.

5. 5 Simplify a numerical expression containing a rational exponent with a numerator of 1 We have seen that a positive integer exponent indicates the number of times that a base is to be used as a factor in a product. For example, x 4 means that x is to be used as a factor four times. x 4 = x  x  x  x 4 factors of x

6. 6 Simplify a numerical expression containing a rational exponent with a numerator of 1 Furthermore, we recall the following rules of exponents. If m and n are natural numbers and x, y  0, then x m x n = x m + n (x m ) n = x mn (xy) n = x n y n x 0 = 1

7. 7 Simplify a numerical expression containing a rational exponent with a numerator of 1 To give meaning to rational (fractional) exponents, we consider. Because is the positive number whose square is 7, we have We now consider the symbol 7 1/2. If fractional exponents are to follow the same rules as integer exponents, the square of 7 1/2 must be 7, because (7 1/2 ) 2 = (7 1/2 ) 2 = 7 1 = 7 Keep the base and multiply the exponents.

8. 8 Simplify a numerical expression containing a rational exponent with a numerator of 1 Since (7 1/2 ) 2 and both equal 7, we define 7 1/2 to be. Similarly, we make these definitions: and so on. Rational Exponents If n is a positive integer greater than 1 and is a real number, then

9. 9 Example Simplify: a. 64 1/2 b. 64 1/3 c. (–64) 1/3 d. 64 1/6 Solution: In each case we will change from rational exponent notation to radical notation and simplify. a. 64 1/2 = = 8 b. 64 1/3 = = 4

10. 10 Example – Solution c. (–64) 1/3 = = –4 d. 64 1/6 = = 2 cont’d

11. 11 Simplify a numerical expression containing a rational exponent with a numerator other than 1 2.

12. 12 Simplify a numerical expression containing a rational exponent with a numerator other than 1 We can extend the definition of x 1/n to cover fractional exponents for which the numerator is not 1. For example, because 4 3/2 can be written as (4 1/2 ) 3, we have 4 3/2 = (4 1/2 ) 3 = = 2 3 = 8 Because 4 3/2 can also be written as (4 3 ) 1/2, we have 4 3/2 = (4 3 ) 1/2 = 64 1/2 = = 8 In general, x m/n can be written as (x 1/n ) m or as (x m ) 1/n.

13. 13 Simplify a numerical expression containing a rational exponent with a numerator other than 1 Since (x 1/n ) m = and (x m ) 1/n =, we make the following definition. Changing from Rational Exponents to Radicals If m and n are positive integers, the fraction m/n cannot be simplified, and x  0 if n is even, then

14. 14 Simplify a numerical expression containing a rational exponent with a numerator other than 1 Comment Recall that the radicand of an even root cannot be negative if we are working with real numbers. In the previous rule, we could state that “x is nonnegative if n is even” to emphasize this fact. There are no such restrictions on odd roots.

15. 15 Simplify: a. 8 2/3 b. (–27) 4/3 Solution: In each case, we will apply the rule. a. or Example

16. 16 b. or cont’d Example – Solution

17. 17 Simplify a numerical expression containing a rational exponent with a numerator other than 1 The work in Example 2 suggests that in order to avoid large numbers, it is usually easier to take the root of the base first.

18. 18 Apply the rules of exponents to an expression containing rational exponents 3.

19. 19 Apply the rules of exponents to an expression containing rational exponents The familiar rules of exponents are valid for rational exponents. The next example illustrates the use of each rule.

20. 20 Example Use the provided rule to write each expression on the left in a different form. Problem Rule a. 4 2/5 4 1/5 = 4 2/5 + 1/5 x m x n = x m + n = 4 3/5 b. (5 2/3 ) 1/2 = 5 (2/3)(1/2) (x m ) n = x mn = 5 1/3 c. (3x) 2/3 = 3 2/3 x 2/3 (xy) m = x m y m

21. 21 Example d. e. f. g. cont’d