✓ Got It?

2. What is each real-number root?
   1. − 27 3 cube root of negative 27 end root ,
   2. − 81 4 the fourth , root of negative 81 end root ,
   3. ( − 7 ) 2 square root of open , negative 7 , close squared end root
   4. − 49 square root of negative 49 end root

Take Note Property: nth Roots of nth Powers

For any real number a, a n n = { a  if  n  is odd | a |  if  n  is even the th , root of eh to the n end root , , equals . left brace . table with 2 rows and 1 column , row1 column 1 , eh , if , n , isodd , row2 column 1 , absolute value of eh , . if , n , iseven , end table

Problem 3 Simplifying Radical Expressions

What is a simpler form of each radical expression?

1. 16 x 8 square root of 16 , x to the eighth end root

   Plan

   How can you get started?

   You're simplifying a square root, so use properties of exponents to write the entire radicand as a perfect square.

   16 x 8 = 4 2 ( x 4 ) 2 = ( 4 x 4 ) 2 = | 4 x 4 | = 4 x 4 square root of 16 , x to the eighth end root . equals . square root of 4 squared . open , x to the fourth , close squared end root . equals . square root of open , 4 , x to the fourth , close squared end root . equals absolute value of , 4 , x to the fourth , end absolute value , . equals 4 , x to the fourth

You need to include absolute value symbols because the index of a square root is 2, which is even. However, | 4 x 4 | = 4 x 4 vertical line , 4 x to the fourth , vertical line equals , 4 x to the fourth  because x 4 x to the fourth  is always nonnegative.

2. a 6 b 9 3 cube root of eh to the sixth , b to the ninth end root ,

   a 6 b 9 3 = ( a 2 ) 3 ( b 3 ) 3 3 = ( a 2 b 3 ) 3 3 = a 2 b 3 cube root of eh to the sixth , b to the ninth end root , . equals . cube root of open , eh squared , close cubed . open , b cubed , close cubed end root , . equals . cube root of open . eh squared , b cubed . close cubed end root , . equals , eh squared , b cubed

The index is odd, so you cannot include absolute value symbols here.

100. x 8 y 12 4 the fourth , root of x to the eighth , y to the twelfth end root ,

     x 8 y 12 4 = ( x 2 ) 4 ( y 3 ) 4 4 = ( x 2 y 3 ) 4 4 = x 2 | y 3 | the fourth , root of x to the eighth , y to the twelfth end root , . equals . the fourth , root of open , x squared , close to the fourth . open , y cubed , close to the fourth end root , . equals . the fourth , root of open . x squared , y cubed . close to the fourth end root , . equals , x squared absolute value of , y cubed , end absolute value ,

The index is even. The absolute value symbols ensure that the root is positive when y 3 y cubed  is negative. Absolute value symbols are not needed for x 2 x squared  since x 2 x squared  is always nonnegative.

✓ Got It?

3. What is the simplified form of each radical expression?
   1. 81 x 4 square root of 81 , x to the fourth end root
   2. a 12 b 15 3 cube root of eh to the twelfth . b to the fifteenth end root ,
   3. x 12 y 16 4 the fourth , root of x to the twelfth . y to the sixteenth end root ,