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.

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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

GUIDED PRACTICE for Examples 4 and 5 Solve the equation. Round the result to two decimal places when appropriate. 13. x 3 = 64 x3x3 = 64 take 3rd root of each side. x = 3 64 Simplify. x = 4 SOLUTION

GUIDED PRACTICE for Examples 4 and x 5 = 512 SOLUTION 1 2 x5x5 = 512 Multiply each side by 2. x5x5 = 1024 take 5th root of each side. x = Simplify. x = 4

GUIDED PRACTICE for Examples 4 and x 2 = 108 SOLUTION 3x23x2 = 108 Divide each side by 3. x2x2 = 36 Simplify. x – = + 6 take 2nd root of each side. x = 2 36

GUIDED PRACTICE for Examples 4 and x 3 = 2 SOLUTION 1 4 x3x3 = 2 Multiply each side by 4. x3x3 = 8 take 3rd root of each side. x = 3 8 Simplify. x= 2

GUIDED PRACTICE for Examples 4 and 5 17.( x – 2 ) 3 = –14 SOLUTION ( x – 2 ) 3 = –14 take 3rd root of each side. ( x – 2 )= 3 –14 add 2 to both sides. x = 3 – Write solution. x = 3 – x = – 0.41 Use a calculator.

GUIDED PRACTICE for Examples 4 and 5 18.( x + 5 ) 4 = 16 SOLUTION ( x + 5 ) 4 = 16 take 4th root of each side. ( x + 5 ) = add 5 to each side. x = – 5 Write solutions separately. x = 2 – 5 or x= – 2 – 5 Use a calculator. x = – 3 or x = –7

EXAMPLE 1 Use properties of exponents = (7 1/3 ) 2 = 12 –1 Use the properties of rational exponents to simplify the expression. b. (6 1/2 4 1/3 ) 2 = (6 1/2 ) 2 (4 1/3 ) 2 = /3 = 6 4 2/3 = 6( 1/2 2 ) 4( 1/3 2 ) e.e. 42 1/ /3 = 7( 1/3 2 )= 7 2/3 a. 7 1/4 7 1/2 = 7 (1/4 + 1/2) = 7 3/4 = 12 [5 (–1/5)] c. ( ) –1/5 = [(4 3) 5 ] –1/5 = (12 5 ) –1/ = d /3 = 5 (1 – 1/3) = 5 2/ /3 = = /3 2

GUIDED PRACTICE for Examples 1 and 2 1. (5 1/3 7 1/4 ) 3 =(5 1/3 ) 3 (7 1/4 ) /4 2 1/ /4 = 3 (1 – 1/4) Use the properties of rational exponents to simplify the expression. 20 1/ /2 = /2 3 (4 1/2 ) 3 = 5 1/ /4 3 = /4 =5 7 3/4 = 2 5/4 =2 (3/4 + 1/2) = /4 = 3 3/4 = (2 2 ) 3/2 = 8 =