3.1 Evaluate nth Roots and Use Rational Exponents

1. 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?

2. 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

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. 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(s) of a. a. n = 3, a = –216 b. n = 4, a = 81 SOLUTION

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

5. Rational Exponents Let a1/n be an nth root of a, and let m be a positive integer. See page 167 for KEY CONCEPT

6. 1 1 23 323/5 64 ( )3 = (161/2)3 = 43 = 43 = 64 = = 16  1 1 1 1 1 1 = = = = (321/5)3 323/5 ( )3  5 8 23 8 32 = = = = Evaluate (a) 163/2 and (b)32–3/5. SOLUTION Radical Form Rational Exponent Form a. 163/2 163/2 b. 32–3/5 32–3/5

7. Evaluate the expression with Rational Exponents • 93/2 • 32-2/5

8. Keystrokes Expression Display 7 3 4 9 1 5 12 3 8 7  c. ( 4 )3 = 73/4 Approximate roots with a calculator a. 91/5 1.551845574 1 5 b. 123/8 2.539176951 3 8 4.303517071 4 3

9. Using a calculator to approximate a root Rewrite the problem as 53/4 and enter using ^ or yx key for the exponent.

10. Keystrokes Expression Display 4 2 5 1 64 - 2 3 10. 64 2/3 – 11. (4√ 16)5 –30 2 3 12. (3√–30)2 16 5 4 Evaluate the expression using a calculator. Round the result to two decimal places when appropriate. 9. 42/5 1.74 0.06 32 9.65

11. Solve the equation using nth roots. • 2x4 = 162 x4 = 81 x4 = 34 x = ±3 • (x − 2)3 = 10 x ≈ 4.15

12. x5 = 512 1 1 x5 = 512 2 2 x5 = 1024 x = 5 1024 x = 4 SOLUTION Multiply each side by 2. take 5th root of each side. Simplify.

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

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

15. 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.

16. 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=.

17. 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)