Concept Byte: Powers of Powers and Powers of Products

Use With Lesson 7-4

ACTIVITY

You can use patterns to find a shortcut for simplifying a power raised to a power or a product raised to a power.

Activity 1

Copy and complete each statement in Exercises 1–6.

1. ( 4 5 ) 2 = 4 5 · 4 5 = 4   +   = 4 5 ·   = 4   open , 4 to the fifth , close squared . equals , 4 to the fifth , middle dot , 4 to the fifth , equals . 4 super begin box , , end box plus begin box , , end box end super . equals . 4 super 5 middle dot begin box , , end box end super . equals , 4 super begin box , , end box end super
2. ( 3 6 ) 3 = 3 6 · 3 6 · 3 6 = 3   +   +   = 3 6 ·   = 3   open , 3 to the sixth , close cubed . equals , 3 to the sixth , middle dot , 3 to the sixth , middle dot , 3 to the sixth , equals . 3 super begin box , , end box plus begin box , , end box plus begin box , , end box end super . equals . 3 super 6 middle dot begin box , , end box end super . equals , 3 super begin box , , end box end super
3. ( 5 8 ) 4 = 5 8 · 5 8 · 5 8 · 5 8 = 5   +   +   +   = 5 8 ·   = 5   open , 5 to the eighth , close to the fourth . equals , 5 to the eighth , middle dot , 5 to the eighth , middle dot , 5 to the eighth , middle dot , 5 to the eighth , equals . 5 super begin box , , end box plus begin box , , end box plus begin box , , end box plus begin box , , end box end super . equals . 5 super 8 middle dot begin box , , end box end super . equals , 5 super begin box , , end box end super
4. ( a 4 ) 2 = a 4 · a 4 = a   +   = a 4 ·   = a   open , eh to the fourth , close squared . equals , eh to the fourth , middle dot , eh to the fourth , equals . eh super begin box , , end box plus begin box , , end box end super . equals . eh super 4 middle dot begin box , , end box end super . equals , eh super begin box , , end box end super
5. ( n 2 ) 3 =   ·   ·   = n   +   +   = n 2 ·   = n   open , n squared , close cubed . equals begin box , , end box middle dot begin box , , end box middle dot begin box , , end box equals . n super begin box , , end box plus begin box , , end box plus begin box , , end box end super . equals . n super 2 middle dot begin box , , end box end super . equals , n super begin box , , end box end super
6. ( x 5 ) 4 =   ·   ·   ·   = x   +   +   +   = x 5 ·   = x   open , x to the fifth , close to the fourth . equals begin box , , end box middle dot begin box , , end box middle dot begin box , , end box middle dot begin box , , end box equals . x super begin box , , end box plus begin box , , end box plus begin box , , end box plus begin box , , end box end super . equals . x super 5 middle dot begin box , , end box end super . equals , x super begin box , , end box end super
7. 1. Look for a Pattern What pattern do you see in your answers to Exercises 1–6?
   2. Predict Use your pattern to simplify ( y 11 ) 33 . open , y to the eleventh , close to the thirty third . .

Activity 2

Copy and complete each statement in Exercises 8–12.

8. ( 3 n ) 2 = 3 n · 3 n = ( 3 · 3 ) ( n · n ) = 3   n   open 3 n close squared . equals 3 n middle dot 3 n equals open 3 middle dot 3 close open n middle dot n close equals , 3 super begin box , , end box end super . n super begin box , , end box end super
9. ( 2 x ) 3 = 2 x · 2 x · 2 x = ( 2 · 2 · 2 ) ( x · x · x ) = 2   x   open 2 x close cubed . equals 2 x middle dot 2 x middle dot 2 x equals open 2 middle dot 2 middle dot 2 close open x middle dot x middle dot x close equals , 2 super begin box , , end box end super . x super begin box , , end box end super
10. ( a b ) 2 = a b · a b = ( a · a ) ( b · b ) = a   b   open eh b close squared . equals eh b middle dot eh b equals open eh middle dot eh close open b middle dot b close equals , eh super begin box , , end box end super . b super begin box , , end box end super
11. ( x y ) 3 = x y · x y · x y = (   ·   ·   ) (   ·   ·   ) = x   y   open x y close cubed . equals x y middle dot x y middle dot x y equals open begin box , , end box middle dot begin box , , end box middle dot begin box , , end box close open begin box , , end box middle dot begin box , , end box middle dot begin box , , end box close equals , x super begin box , , end box end super . y super begin box , , end box end super
12. ( p q ) 4 =   ·   ·   ·   = (   ·   ·   ·   ) (   ·   ·   ·   ) = p   q   open p q close to the fourth . equals begin box , , end box middle dot begin box , , end box middle dot begin box , , end box middle dot begin box , , end box equals open begin box , , end box middle dot begin box , , end box middle dot begin box , , end box middle dot begin box , , end box close open begin box , , end box middle dot begin box , , end box middle dot begin box , , end box middle dot begin box , , end box close equals , p super begin box , , end box end super . q super begin box , , end box end super
13. 1. Look for a Pattern What pattern do you see in your answers to Exercises 8 − 12 ? 8 minus 12 question mark
    2. Predict Use your pattern to simplify ( r s ) 20 . open r s close to the twentieth . .

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