1. WARM UP 1.Multiply and simplify 2. Rationalize the denominator 3. Rationalize the denominator 1

2. RATIONAL NUMBERS AS EXPONENTS

3. OBJECTIVES Calculate radical expressions of the form in two ways. 3 Write expressions with rational exponents as radical expressions and vice versa. Simplify expressions containing negative rational exponents. Use rational exponents to simplify radical expressions. Simplify more-involved radical expressions when solving algebraic equations and problems.

4. EXPONENTS WITH RADICALS 4 Simplify and compare your answers and The results of the introductory work suggest the following theorem: Theorem 7-6 For any nonnegative number a, any natural number index k, and any integer m, and

5. COMBINATION OF POWERS AND ROOTS 5 We can raise to a power and then take a root, or we can take a root and then raise to a power. One method of simplifying may be easier than the other. Examples: 1. 2. 3. 4. or

6. TRY THIS… Simplify as shown. Then use Theorem 7-6 to simplify another way. a. b. 6

7. RATIONAL EXPONENTS To extend the idea of an exponent to include rational exponents, consider. If the usual properties of exponents are to hold, then should equal or a. 7 We also know that. Thus should be defined as. Similarly, or a. Thus should be defined as DEFINITION For any nonnegative number a and any natural number k, means (the nonnegative kth root of a)

8. EXAMPLES When working with rational exponents, we will assume that variables in the base are nonnegative. 8 Write without rational exponents: Write with rational exponents:

9. TRY THIS… Write without rational exponents a. b. c. Write with rational exponents d.e. or 2

10. MORE RATIONAL EXPONENTS How should we define ? If the usual properties of exponents are to hold, we have or or 10 DEFINITION For any natural numbers m and k, and any nonnegative number a, means Thus, represents the principal kth root of. Since by Theorem 7-6 we know that, it follows that also represents.

11. EXAMPLES Write without rational exponents 11 or Write with rational exponents

12. TRY THIS… Write without rational exponents a. b. Write with rational exponents c.d. e.

13. NEGATIVE RATIONAL EXPRESSION Negative rational exponents have a meaning similar to that of negative integer exponents. 13 DEFINITION For any rational number and any positive real number a, means Changing the sign of an exponent amounts to finding a reciprocal. and are reciprocals.

14. EXAMPLES Rewrite with positive exponents. 14 is the reciprocal of Since the answer simplifies to is the reciprocal of

15. TRY THIS… Rewrite with positive exponents: a. b. 15

16. EXAMPLES The properties of exponents that hold for integer exponents also hold for rational exponents. 16 Adding exponents Use properties of exponents to simplify: Subtracting exponents Multiplying exponents

17. TRY THIS… Use properties of exponents to simplify: a. b. c. 17

18. SIMPLIFYING RADICAL EXPRESSIONS Rational exponents can be used to simplify some radical expressions. The procedure is a follows: 18 Converting to an exponential expression 1. Convert radical expressions to exponential expressions. Simplifying the exponent Converting back to radical notation 2. Use the properties of exponents to simplify. 3. Convert back to radical notation when appropriate. Examples: Use rational exponents to simplify.

19. TRY THIS… Use rational exponents to simplify: a. b. c.

20. MORE EXAMPLES Use rational exponents to simply 20

21. TRY THIS… Use rational exponents to simplify: a. b.

22. MORE EXAMPLES We can use properties of rational exponents to write a single radical expression for a product or quotient. 22 Examples: Write as a single radical expression

23. MORE EXAMPLES 23 Examples: Write as a single radical expression

24. RATIONAL RADICAL EXAMPLES 24 Examples: Write as a single radical expressions

25. TRY THIS… Write as a radical expression a. b. 25

26. RADICAL EXPRESSIONS Write as a single radical expression 26 Rewriting exponents with a common denominator. Using the properties of exponents Converting to radical notation

27. TRY THIS… Write as a single radical expression a. b.

28. SIMPLIFYING RADICAL EXPRESSIONS We now have seen four different methods of simplifying radical expressions. 28 1.Simplifying by factoring: Factor the radicand, looking for factors that are perfect powers. 2.Rationalizing denominators: Multiply the radical expression by 1 to make the denominator a perfect square. Then simplify the expression. 3.Collecting like radical terms: Use the distributive laws to collect terms with the same radicand. 4.Using rational exponents: Convert to exponential notation; use the properties of exponents to simplify. Convert back to radical notation.

29. CH. 7.5 HOMEWORK Textbook pg. 315 #2, 6, 10, 16, 22, 26, 28, 34, 42, 46, 52, & 54 www.brainybetty.com29