8-6 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz

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1 8-6 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt Algebra ALgebra2 2

2 Warm Up Simplify each expression , (3 2 )

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

4 index rational exponent Vocabulary

5 You are probably familiar with finding the square root of a number. These two operations are inverses of each other. Similarly, there are roots that correspond to larger powers. 5 and 5 are square roots of 25 because 5 2 = 25 and ( 5) 2 = 25 2 is the cube root of 8 because 2 3 = 8. 2 and 2 are fourth roots of 16 because 2 4 = 16 and ( 2) 4 = 16. a is the nth root of b if a n = b.

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 Find all real roots. Example 1: Finding Real Roots A. sixth roots of 64 A positive number has two real sixth roots. Because 2 6 = 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 Find all real roots. a. fourth roots of 256 Check It Out! Example 1 A negative number has no real fourth roots. b. sixth roots of 1 A positive number has two real sixth roots. Because 1 6 = 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.

10 The properties of square roots in Lesson 1-3 also apply to nth roots.

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. 3 x x x 3x 3 Product Property. Simplify.

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

14 Check It Out! Example 2a Simplify the expression. Assume that all variables are positive x x 4 Factor into perfect fourths x 4 2 x 2x Product Property. Simplify.

15 Check It Out! Example 2b Simplify the expression. Assume that all variables are positive. 4 x Quotient Property. Rationalize the numerator. Product Property x 2 Simplify.

16 Check It Out! Example 2c Simplify the expression. Assume that all variables are positive. x g x x x 3 Product Property of Roots. Simplify.

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

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

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

20 Write the expression simplify. Check It Out! Example 3a in radical form, and Method 1 Evaluate the root first. Method 2 Evaluate the power first. ( ) 1 Write with a radical ( ) Write will a radical. (4) 1 Evaluate the root Evaluate the power. 4 Evaluate the power. 4 Evaluate the root.

21 Write the expression simplify. Method 1 Evaluate the root first. ( ) 5 Check It Out! Example 3b Write with a radical. 2 4 ( ) 5 in radical form, and Method 2 Evaluate the power first. 2 4 Write with a radical. (2) 5 Evaluate the root Evaluate the power. 32 Evaluate the power. 32 Evaluate the root.

22 Check It Out! Example 3c Write the expression in radical form, and simplify. Method 1 Evaluate the root first. Method 2 Evaluate the power first. ( ) 3 Write with a radical ( ) Write with a radical. (5) 3 Evaluate the root ,140,625 Evaluate the power. 125 Evaluate the power. 125 Evaluate the root.

23 Example 4: Writing Expressions by Using Rational Exponents Write each expression by using rational exponents. A. B n m m = n n m a a a = a m n 13 1 Simplify. 3 3 Simplify. 27

24 Check It Out! Example 4 Write each expression by using rational exponents. a. b. c Simplify Simplify.

25 Rational exponents have the same properties as integer exponents (See Lesson 1-5)

26 Example 5A: Simplifying Expressions with Rational Exponents Simplify each expression. Product of Powers Simplify. Evaluate the Power. Check Enter the expression in a graphing calculator.

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

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

29 Check It Out! Example 5a Simplify each expression. Product of Powers. Simplify. 6 Evaluate the Power. Check Enter the expression in a graphing calculator.

30 Check It Out! Example 5b Simplify each expression. ( 8) Negative Exponent Property. 1 2 Evaluate the Power. Check Enter the expression in a graphing calculator.

31 Check It Out! Example 5c Simplify each expression Quotient of Powers. Simplify. Evaluate the power. Check Enter the expression in a graphing calculator.

32 Example 6: Chemistry Application Radium-226 is a form of radioactive element that decays over time. An initial sample of radium-226 has a mass of 500 mg. The mass of radium-226 remaining from the initial sample after t years is given by. To the nearest milligram, how much radium-226 would be left after 800 years?

33 t Example 6 Continued (200 ) = 500( ) Substitute 800 for t. 1 2 = 500(2 ) 1 = 500( ) = The amount of radium-226 left after 800 years would be about 354 mg. Simplify. Negative Exponent Property. Simplify. Use a calculator.

34 Check It Out! Music Application To find the distance a fret should be place from the bridge on a guitar, multiply the length of the string by, where n is the number of notes higher that the string s root note. Where should the fret be placed to produce the E note that is one octave higher on the E string (12 notes higher)? = 64(2 1 ) = 32 Use 64 cm for the length of the string, and substitute 12 for n. Simplify. Negative Exponent Property. Simplify. The fret should be placed 32 cm from the bridge.

35 Find all real roots. Lesson Quiz: Part I 1. fourth roots of 625 5, 5 2. fifth roots of Simplify each expression y 4y Write ( 216) in radical form and simplify. ( - ) = Write 21 using rational exponents. 21 3

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? $

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