04 Evaluate nth Roots and Use Rational Exponents

04 Evaluate nth Roots and Use Rational Exponents pg. P 14 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. 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.

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

Rational Exponents Let a 1/n be an nth root of a, and let m be a positive integer. Powers go up, Roots go down

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 = 43 = 64 32– 3/5 1 1 = = 3/5 5 32 ( 32 )3 = 1 2 8

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

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 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 Display 4 2 5 1. 74 1 (64 2 3 0. 06 16 5 4 32 – 30 2 3 9. 65

Solve the equation using nth roots. 2 x 4 = 162 x 4 = 81 x 4 = 34 x = ± 3 (x − 2)3 = 10 x ≈ 4. 15

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

( x – 2 )3 = – 14 SOLUTION ( x – 2 )3= – 14 (x– 2)= 3 – 14 x = 3 – 14 + 2 x = – 0. 41 Use a calculator.

( x + 5 )4 = 16 SOLUTION ( x + 5 )4 = 16 (x+5) =+ x =+ 4 4 16 16 – 5 take 4 th 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 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=.