5-6 Radical Expressions and Rational Exponents

Transcription

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

2 Warm Up Simplify each expression (3 2 ) 3 16,

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 Essential Question How do you multiply/divide powers with the same base when the exponents are rational?

6 You are probably familiar with finding the square root of a number and squaring 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 = 16 and ( 2) = 16. a is the nth root of b if a n = b.

7 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 unless otherwise stated. Reading Math When a radical sign shows no index, it represents a square root (index of 2).

8 Find all real roots. Example 1: Finding Real Roots A. sixth roots of 6 A positive number has two real sixth (even) roots. Because 2 6 = 6 and ( 2) 6 = 6, the roots are 2 and 2. B. cube roots of 216 A negative number has one real cube (odd) root. Because ( 6) 3 = 216, the root is 6. C. fourth roots of 102 A negative number has no real fourth (even) roots.

9 The properties of square roots in Lesson 2-5 also apply to nth roots.

10 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.

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

12 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. The denominator of a rational exponent becomes the index of the radical.

13 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.

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

15 Example : 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 Simplify. Simplify

16 Check It Out! Example Write each expression by using rational exponents. a. b. 3 3 ( 3 ) c ( 3 ) 3 3 Simplify Simplify Simplify. 27

17 Rational exponents have the same properties as integer exponents.

18 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.

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

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

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

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

23 Find all real roots. Lesson Quiz: Part I 1. fourth roots of 625 5, 5 2. fifth roots of 23 3 Simplify each expression y 8 y Write ( 216) in radical form and simplify. ( )= Write 21 using rational exponents. 21 3

24 Essential Question How do you multiply/divide powers with the same base when the exponents are rational?

25 Q: How is an artificial tree like the fourth root of 68? A: Neither has real roots.

26 Find all real roots. a. fourth roots of 256 A negative number has no real fourth roots. b. sixth roots of 1 Check It Out! Example 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.

27 Check It Out! Example 2a Simplify the expression. Assume that all variables are positive. 16x 2 x Factor into perfect fourths. 2 x 2 x 2x Product Property. Simplify.

28 Check It Out! Example 2b Simplify the expression. Assume that all variables are positive. x 8 3 Quotient Property. Rationalize the numerator. Product Property x 2 Simplify.

29 Check It Out! Example 3a Write the expression simplify in radical form, and Method 1 Evaluate the root first. Method 2 Evaluate the power first. ( ) 1 Write with a radical. 3 6 ( ) Write will a radical. () 1 Evaluate the root. 3 6 Evaluate the power. Evaluate the power. Evaluate the root.

30 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. 625 ( ) Write with a radical. (5) 3 Evaluate the root. 2,10,625 Evaluate the power. 125 Evaluate the power. 125 Evaluate the root.