Problem 4 Simplifying an Exponential Expression

What is the simplified form of ( 2 x 6 y 4 ) − 3 . open . fraction 2 , x to the sixth , over y to the fourth end fraction . close super negative 3 end super . .

( 2 x 6 y 4 ) − 3 = ( y 4 2 x 6 ) 3 Rewrite using the reciprocal of  2 x 6 y 4 .   = ( y 4 ) 3 ( 2 x 6 ) 3 Raise the numerator and denominator to the third power .   = y 12 8 x 18 Simplify . table with 3 rows and 3 columns , row1 column 1 , open . fraction 2 , x to the sixth , over y to the fourth end fraction . close super negative 3 end super , column 2 equals . open . fraction y to the fourth , over 2 , x to the sixth end fraction . close cubed , column 3 cap rewriteusingthereciprocalof . fraction 2 , x to the sixth , over y to the fourth end fraction . . , row2 column 1 , , column 2 equals . fraction open , y to the fourth , close cubed , over open , 2 , x to the sixth , close cubed end fraction , column 3 cap raisethenumeratoranddenominatortothethirdpower . . , row3 column 1 , , column 2 equals . fraction y to the twelfth , over 8 , x to the eighteenth end fraction , column 3 cap simplify , . , end table

✓ Got It?

4. What is the simplified form of ( a 5 b ) − 2 ? open , fraction eh , over 5 b end fraction , close super negative 2 end super . question mark

✓ Lesson Check

Do you know HOW?

Simplify each expression.

1. y 3 y 10 fraction y cubed , over y to the tenth end fraction
2. ( x 4 3 ) 3 open , fraction x to the fourth , over 3 end fraction , close cubed
3. ( m n ) − 3 open , m over n , close super negative 3 end super
4. ( 3 x 2 5 y 4 ) − 4 open . fraction 3 super x 2 end super , over 5 super y 4 end super end fraction . close super negative 4 end super
5. A large cube is made up of many small cubes. The volume of the large cube is 7.506 × 10 5 mm 3 . 7.506 , times , 10 to the fifth . mm cubed , .  The volume of each small cube is 2.78 × 10 4 mm 3 . 2.78 , times , 10 to the fourth . mm cubed , .  How many small cubes make up the large cube?

Do you UNDERSTAND?

6. Vocabulary How is the property for raising a quotient to a power similar to the property for raising a product to a power?

   1. Reasoning Ross simplifies

      a 3 a 7 fraction eh cubed , over eh to the seventh end fraction  as shown below. Explain why Ross's method works.

      a 3 a 7 = 1 a 7 − 3 = 1 a 4 fraction eh cubed , over eh to the seventh end fraction . equals . fraction 1 , over eh super 7 minus 3 end super end fraction . equals , fraction 1 , over eh to the fourth end fraction

   2. Open-Ended Write a quotient of powers and use Ross's method to simplify it.