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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? 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. n a a1 n If n is even and a > 0, then a has two real nth roots. a a n 1n If n is even and a = 0, then a has one nth root. n 0 0 1n 0 If n is even and a < o, then a has no real nth roots. See page 166 for KEY CONCEPT Find the indicated real nth root(s) of a. a. n = 3, a = –216 b. n = 4, a = 81 SOLUTION 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. b. Because n = 4 is even and a = 81 > 0, 81 has two real fourth roots. Because 34 = 81 and (–3)4 = 81, you can write ±4√ 81 = ±3 Find the indicated real nth root n = 3, a = −125 n = 4, a = 16 3 125 3 16 4 3 5 5 4 2 2 4 Rational Exponents Let a1/n be an nth root of a, and let m be a positive integer. a mn a a a m n a 1n m n 1 1 mn a 1n m m 1 a n m ,a 0 See page 167 for KEY CONCEPT Evaluate (a) 163/2 and (b) 32–3/5. SOLUTION Rational Exponent Form a. 163/2= (161/2)3= 43 = 64 b. 1 1 = 323/5 (321/5)3 1 = 1 = 3 8 2 32–3/5= Radical Form 3 3/2 ( ) 16 = 16 = 43 = 64 32– 3/5 1 1 = 3/5 = 5 32 ( 32 )3 = 13 = 1 2 8 Evaluate the expression with Rational Exponents 93/2 32-2/5 9 3 3 1 25 32 3 1 32 5 2 27 1 2 5 25 1 1 2 2 4 Approximate roots with a calculator Keystrokes Expression a. 91/5 b. 123/8 c. ( 4 7)3 = 73/4 Display 9 11 5 5 1.551845574 12 3 3 8 8 2.539176951 7 3 3 4 4 4.303517071 Using a calculator to approximate a root 5 4 3 3.34 Rewrite the problem as 53/4 and enter using ^ or yx key for the exponent. Evaluate the expression using a calculator. Round the result to two decimal places when appropriate. Keystrokes Expression 9. 42/5 10. 64 –2/3 11. (4√ 16)5 12. (3√ –30)2 4 1 64 Display 2 5 1.74 -2 3 0.06 16 5 4 32 –30 2 3 9.65 Solve the equation using nth roots. 2x4 = 162 x4 = 81 x4 = 34 x = ±3 (x − 2)3 = 10 x 2 3 10 3 x 10 2 x ≈ 4.15 1 x5 = 512 2 SOLUTION 1 x5 = 512 2 x5 = 1024 x = 5 x = 4 1024 Multiply each side by 2. take 5th root of each side. Simplify. ( x – 2 )3 = –14 SOLUTION ( x – 2 )3= –14 (x–2)= 3 –14 x = 3 –14 + 2 x = 3 –14 + 2 x = – 0.41 Use a calculator. ( x + 5 )4 = 16 SOLUTION ( x + 5 )4 = 16 4 (x+5) =+ x =+ 4 16 16 – 5 take 4th root of each side. add 5 to each side. x = 2 – 5 or x = – 2 – 5 Write solutions separately. x = –3 x = –7 Use a calculator. or 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. 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=. Assignment Page 169, 9-45 every 3rd 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)