EXAMPLE 1 Find nth roots Find the indicated real nth root ( s ) of a. a. n = 3, a = –216 b. 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.

EXAMPLE 2 Evaluate expressions with rational exponents 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  = = 64= b. 32 –3/5 = /5 = 1 (32 1/5 ) 3 = = 32 –3/ /5 = 1 ( ) 3 5  32 = = =

EXAMPLE 3 Approximate roots with a calculator Expression Keystrokes Display a. 9 1/ b. 12 3/ c. ( 4 ) 3 = 7 3/4 

EXAMPLE 4 Solve equations using nth roots Solve the equation. a. 4x 5 = 128 Divide each side by 4. x5x5 32= Take fifth root of each side. x=  32 5 Simplify. x2 =

EXAMPLE 4 Solve equations using nth roots b. (x – 3) 4 = 21 Take fourth roots of each side. x – –  21= Add 3 to each side. x  4 + – 21+ 3= Write solutions separately. x 4  = or x= 4  21+ 3– Use a calculator. x5.14 or x0.86

EXAMPLE 5 Use nth roots in problem solving Biology A study determined that the weight w ( in grams ) of coral cod near Palawan Island, Philippines, can be approximated using the model w = ℓ 3 where l is the coral cod’s length ( in centimeters ). Estimate the length of a coral cod that weighs 200 grams.

EXAMPLE 5 Use nth roots in problem solving SOLUTION Write model for weight. w l 3 = Substitute 200 for w l 3 = Divide each side by ,976 l 3 Take cube root of each side. 11,976 3  l Use a calculator l ANSWER A coral cod that weighs 200 grams is about 23 centimeters long.