04 Evaluate nth Roots and Use Rational Exponents pg. P14 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 a1/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 Radical Form a. 163/2 (161/2)3 = 43 = 64 = 163/2 ( )3  = 16 43 = 64 = = 1 323/5 = 1 (321/5)3 1 323/5 = 1 ( )3 5  32 = b. 32–3/5 32–3/5 = 1 23 1 8 = = 1 23 1 8 =

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

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

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

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

1 2 x5 = 512 SOLUTION 1 2 x5 = 512 x5 = 1024 Multiply each side by 2. x = 5 1024 take 5th root of each side. x = 4 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 ( x + 5 ) = + 4 16 take 4th root of each side. add 5 to each side. x = + 4 16 – 5 x = 2 – 5 or = – 2 – 5 Write solutions separately. x = – 3 or = –7 Use a calculator.

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=.