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3.1 Evaluate nth Roots and Use Rational Exponents p. 166 What is a quick way to tell what kind of real roots you have? How do you write a radical in exponent form? What buttons do you use on a calculator to approximate a radical? What is the difference between evaluating and solving?

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Real nth Roots Let n be an integer greater than 1 and a be a real number. If n is odd, then a has one real nth root. If n is even and a > 0, then a has two real nth roots. If n is even and a = 0, then a has one nth root. If n is even and a < o, then a has no real nth roots. See page 166 for KEY CONCEPT

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Find the indicated real nth root(s) of a. a. n = 3, a = –216b. n = 4, a = 81 SOLUTION b. Because n = 4 is even and a = 81 > 0, 81 has two real fourth roots. Because 3 4 = 81 and (–3) 4 = 81, you can write ± 4 √ 81 = ±3 a. Because n = 3 is odd and a = –216 < 0, –216 has one real cube root. Because (–6) 3 = –216, you can write = 3 √ –216 = –6 or (–216) 1/3 = –6.

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Find the indicated real nth root n = 3, a = −125 n = 4, a = 16

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Rational Exponents Let a 1/n be an nth root of a, and let m be a positive integer. See page 167 for KEY CONCEPT

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Evaluate (a) 16 3/2 and (b) 32 –3/5. SOLUTION Rational Exponent Form Radical Form a. 16 3/2 (16 1/2 ) 3 = 4343 = 64 = 16 3/2 ( ) 3 = 16 4 3 = 64= b. 32 –3/5 = 1 32 3/5 = 1 (32 1/5 ) 3 = 1 2323 1 8 = 32 – 3/5 1 32 3/5 = 1 ( ) 3 5 32 = = 1 2323 1 8 =

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Evaluate the expression with Rational Exponents 9 3/2 32 -2/5

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Approximate roots with a calculator Expression Keystrokes Display a. 9 1/5 9 1 51.551845574 b. 12 3/8 12 3 82.539176951 7 c. ( 4 ) 3 = 7 3/4 7 3 44.303517071 1 5 8 3 3 4

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Using a calculator to approximate a root Rewrite the problem as 5 3/4 and enter using ^ or y x key for the exponent.

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Expression Keystrokes Display 9. 4 2/5 4 2 51.74 - 2 3 1 64 0.06 16 5 432 –30 2 39.65 10. 64 2/3 – 11. ( 4 √ 16) 5 12. ( 3 √ –30) 2 Evaluate the expression using a calculator. Round the result to two decimal places when appropriate.

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Solve the equation using nth roots. 2x 4 = 162 x 4 = 81 x 4 = 34 x = ±3 (x − 2) 3 = 10 x ≈ 4.15

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1 2 x 5 = 512 SOLUTION 1 2 x5x5 = 512 Multiply each side by 2. x5x5 = 1024 take 5th root of each side. x = 5 1024 Simplify. x = 4

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( x – 2 ) 3 = –14 SOLUTION ( x – 2 ) 3 = –14 ( x – 2 )= 3 –14 x = 3 –14 + 2 x x = – 0.41 Use a calculator.

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( x + 5 ) 4 = 16 SOLUTION ( x + 5 ) 4 = 16 take 4th root of each side. ( x + 5 ) = + 4 16 add 5 to each side. x = + 4 16 – 5 Write solutions separately. x = 2 – 5 or x= – 2 – 5 Use a calculator. x = – 3 orx = –7

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Evaluating a model with roots. When you take a number to with a rational exponent and express it in an integer answer, you have evaluated. Solving an equation using an nth root. When you have an equation with value that has a rational exponent, you solve the equation to find the value of the variable.

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What is a quick way to tell what kind of real roots you have? Root is odd, 1 answer; root is even, 1 or 2 real answers. How do you write a radical in exponent form? Use a fraction exponent (powers go up, roots go down) What buttons do you use on a calculator to approximate a radical? Root buttons What is the difference between evaluating and solving? Evaluating simplifies; Solving finds answers x=.

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Assignment Page 169, 9-45 every 3 rd problem, 50-56 even, To get credit for doing the problem, you must show the original problem along with your answer unless it is a calculator problem (41-51)

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