C 5 1 4 ⋅ 125 1 4 5 super and 1 fourth end super , dot . 125 super and 1 fourth end super

5 1 4 ⋅ 125 1 4 = 5 4 ⋅ 125 4 Rewrite as radicals .   = 5 ⋅ 125 4 Property for multiplying radical expressions   = 625 4 Multiply .   = 5 4 4 Rewrite the radicand .   = 5 Simplify . table with 5 rows and 3 columns , row1 column 1 , 5 super and 1 fourth end super , dot . 125 super and 1 fourth end super , column 2 equals , the fourth , root of 5 , , dot , the fourth , root of 125 , , column 3 cap rewriteasradicals . . , row2 column 1 , , column 2 equals . the fourth , root of 5 dot 125 end root , , column 3 cap propertyformultiplyingradicalexpressions , row3 column 1 , , column 2 equals , the fourth , root of 625 , , column 3 cap multiply , . , row4 column 1 , , column 2 equals , the fourth , root of 5 to the fourth end root , , column 3 cap rewritetheradicand . . , row5 column 1 , , column 2 equals 5 , column 3 cap simplify , . , end table

✓ Got It?

1. What is the simplified form of each expression?
   1. 64 1 2 64 super and 1 half end super
   2. 11 1 2 ⋅ 11 1 2 11 super and 1 half end super , dot , 11 super and 1 half end super
   3. 3 1 2 ⋅ 12 1 2 3 super and 1 half end super , dot , 12 super and 1 half end super

Take Note Key Concept: Rational Exponent

If the nth root of a is a real number, m is an integer, and m n m over n  is in lowest terms, then a 1 n = a n  and  a m n = a m n = ( a n ) m . eh super 1 over n end super , equals , the th , root of eh , , and . eh super m over n end super , equals , the th , root of eh to the m end root , , equals . open , the th , root of eh , , close to the m . .  If m is negative, a ≠ 0 . eh not equal to 0 .

Problem 2 Converting Between Exponential and Radical Form

A What are x 3 7 x super 3 sevenths end super  and y − 3.5 y super negative 3.5 end super  in radical form?

x 3 7 = x 3 7  or  ( x 7 ) 3 y − 3.5 = y − 7 2   = y 1 7 2   = 1 y 7 = 1 y 6 y = 1 y 3 y  or  y y 4 table with 2 rows and 1 column , row1 column 1 , x super 3 sevenths end super , equals , the seventh , root of x cubed end root , . or . open , the seventh , root of x , , close cubed , row2 column 1 , table with 3 rows and 2 columns , row1 column 1 , y super negative 3.5 end super , column 2 equals . y super negative , 7 halves end super , row2 column 1 , , column 2 equals . y super fraction 1 , over and 7 halves end fraction end super , row3 column 1 , , column 2 equals , fraction 1 , over square root of y to the seventh end root end fraction , equals . fraction 1 , over square root of y to the sixth , y end root end fraction . equals . fraction 1 , over y cubed , square root of y end fraction . or . fraction square root of y , over y to the fourth end fraction , end table , end table

B What are a 5 square root of eh to the fifth end root  and ( b 5 ) 3 open , the fifth , root of b , , close cubed  in exponential form?

a 5 = ( a 5 ) 1 2 = a 5 2 ( b 5 ) 3 = ( b 1 5 ) 3 = b 3 5 table with 2 rows and 1 column , row1 column 1 , square root of eh to the fifth end root , equals . open , eh to the fifth , close super 1 half end super . equals , eh super 5 halves end super , row2 column 1 , open , the fifth , root of b , , close cubed . equals . open , b super 1 fifth end super , close cubed . equals , b super 3 fifths end super , end table

✓ Got It?

2. 1. What are the expressions w − 5 8 w super negative , 5 eighths end super  and w 0.2 w to the 0.2  in radical form?
   2. What are the expressions x 3 4 the fourth , root of x cubed end root ,  and ( y 5 ) 4 open , the fifth , root of y , , close to the fourth  in exponential form?
   3. Reasoning Refer to the definition of rational exponent. Explain the need for the restriction that a ≠ 0 eh not equal to 0  if m is negative.