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1. 5-6 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt ALgebra2

2. 1. r= • x= 7 and x=-1 • 4. d= 2. no solution. • x < -1 or x > 0 • -5 < x -3 • -3 < x -2 • x < 3 OR x > 4 • m < 0 or m 4 • 5 < s < 9 • z -24 or z > 4 • x < -12 or x > 15

3. 118 2. 116 75 5 3 2 35 20 7 7 Warm Up Simplify each expression. 16,807 1. 73•72 121 729 3. (32)3 4. 5.

4. Objectives Rewrite radical expressions by using rational exponents. Simplify and evaluate radical expressions and expressions containing rational exponents.

5. Vocabulary index rational exponent

6. The nth root of a real number a can be written as the radical expression , where n is the index (plural: indices) of the radical and a is the radicand. When a number has more than one root, the radical sign indicates only the principal, or positive, root.

7. Reading Math When a radical sign shows no index, it represents a square root.

8. Example 1: Finding Real Roots Find all real roots. A. sixth roots of 64 A positive number has two real sixth roots. Because 26 = 64 and (–2)6 = 64, the roots are 2 and –2. B. cube roots of –216 A negative number has one real cube root. Because (–6)3 = –216, the root is –6. C. fourth roots of –1024 A negative number has no real fourth roots.

9. Check It Out! Example 1 Find all real roots. a. fourth roots of –256 A negative number has no real fourth roots. b. sixth roots of 1 A positive number has two real sixth roots. Because 16 = 1 and (–1)6 = 1, the roots are 1 and –1. c. cube roots of 125 A positive number has one real cube root. Because (5)3 = 125, the root is 5.

11. Remember! When an expression contains a radical in the denominator, you must rationalize the denominator. To do so, rewrite the expression so that the denominator contains no radicals.

12. Example 2A: Simplifying Radical Expressions Simplify each expression. Assume that all variables are positive. Factor into perfect fourths. Product Property. 3  x x x Simplify. 3x3

13. Example 2B: Simplifying Radical Expressions Quotient Property. Simplify the numerator. Rationalize the numerator. Product Property. Simplify.

14. 424 •4 x4 424 •x4 You Try! Example 2a Simplify the expression. Assume that all variables are positive. 4 4 16 x Factor into perfect fourths. Product Property. 2  x Simplify. 2x

15. 8 4 x 4 3 27 4 2 x 3 You Try! Example 2b Simplify the expression. Assume that all variables are positive. Quotient Property. Rationalize the numerator. Product Property. Simplify.

16. 3 9 x You Try! Example 2c Simplify the expression. Assume that all variables are positive. Product Property of Roots. x3 Simplify.

17. Assignment: Simplifying Radical Expressions Worksheet

18. Answers: Simplifying Radical Expressions Worksheet

19. Warm Up

20. A rational exponent is an exponent that can be expressed as , where m and n are integers and n ≠ 0. Radical expressions can be written by using rational exponents. m n

21. Writing Math The denominator of a rational exponent becomes the index of the radical.

22. Write the expression (–32) in radical form and simplify. 3 5 ( ) 3 - 5 32 - 32,768 5 Example 3: Writing Expressions in Radical Form Method 1 Evaluate the root first. Method 2 Evaluate the power first. Write with a radical. Write with a radical. (–2)3 Evaluate the root. Evaluate the power. –8 Evaluate the power. –8 Evaluate the root.

23. 1 3 64 ( ) 1 ( ) 3 64 1 64 3 3 64 You Try! Example 3a Write the expression in radical form, and simplify. Method 1 Evaluate the root first. Method 2 Evaluate the power first. Write with a radical. Write will a radical. (4)1 Evaluate the root. Evaluate the power. 4 Evaluate the power. 4 Evaluate the root.

24. 5 2 4 ( ) 5 ( ) 2 4 5 4 2 2 1024 You Try! Example 3b Write the expression in radical form, and simplify. Method 1 Evaluatethe root first. Method 2 Evaluatethe power first. Write with a radical. Write with a radical. (2)5 Evaluate the root. Evaluate the power. 32 Evaluate the power. 32 Evaluate the root.

25. 3 4 625 ( ) 3 ( ) 3 4 625 625 4 244,140,625 4 You Try! Example 3c Write the expression in radical form, and simplify. Method 1 Evaluatethe root first. Method 2 Evaluate the power first. Write with a radical. Write with a radical. (5)3 Evaluate the root. Evaluate the power. 125 Evaluate the power. 125 Evaluate the root.

26. 4 15 1 2 8 5 A. B. 13 m m = = m m n n a a a a n n 3 13 Example 4: Writing Expressions by Using Rational Exponents Write each expression by using rational exponents. Simplify. 33 Simplify. 27

27. 2 1 3 9 3 2 4 4 5 10 81 5 You Try! Example 4 Write each expression by using rational exponents. a. b. c. 103 Simplify. Simplify. 1000

28. Rational exponents have the same properties as integer exponents

29. Example 5A: Simplifying Expressions with Rational Exponents Simplify each expression. Product of Powers. Simplify. 72 Evaluate the Power. 49 CheckEnter the expression in a graphing calculator.

30. 1 4 Example 5B: Simplifying Expressions with Rational Exponents Simplify each expression. Quotient of Powers. Simplify. Negative Exponent Property. Evaluate the power.

31. Example 5B Continued Check Enter the expression in a graphing calculator.

32. You Try! Example 5a Simplify each expression. Product of Powers. Simplify. 6 Evaluate the Power. Check Enter the expression in a graphing calculator.

33. 1 1 3 3 (–8)– 1 – 1 –8 2 You Try! Example 5b Simplify each expression. Negative Exponent Property. Evaluate the Power. Check Enter the expression in a graphing calculator.

34. You Try! Example 5c Simplify each expression. Quotient of Powers. 52 Simplify. Evaluate the power. 25 Check Enter the expression in a graphing calculator.

35. Assignment: Radical Expressions and Rational Exponents

36. Lesson Quiz: Part II 7. If $2000 is invested at 4% interest compounded monthly, the value of the investment after t years is given by . What is the value of the investment after 3.5 years? $2300.01