7-3 and 7-4 Multiplication Properties of Exponents

Quick Review

To multiply powers with the same base, add the exponents.

a m · a n = a m + n , eh to the m , middle dot , eh to the n , equals . eh super m plus n end super . comma  where a ≠ 0 eh not equal to 0  and m and n are integers

To raise a power to a power, multiply the exponents.

( a m ) n = a m n , open , eh to the m , close to the n . equals eh , m to the n , comma  where a ≠ 0 eh not equal to 0  and m and n are integers

To raise a product to a power, raise each factor in the product to the power.

( a b ) n = a n b n , open eh b close to the n . equals , eh to the n , b to the n , comma  where a ≠ 0 , eh not equal to 0 comma   b ≠ 0 , b not equal to 0 comma  and n is an integer

Example

What is the simplified form of each expression?

1. 3 10 · 3 4 = 3 10 + 4 = 3 14 3 to the tenth , middle dot , 3 to the fourth , equals . 3 super 10 plus 4 end super . equals , 3 to the fourteenth
2. ( x 5 ) 7 = x 5 · 7 = x 35 open , x to the fifth , close to the seventh . equals . x super 5 middle dot 7 end super . equals , x to the thirty fifth
3. ( p q ) 8 = p 8 q 8 open p q close to the eighth . equals , p to the eighth , q to the eighth

Exercises

Complete each equation.

27. 3 2 · 3   = 3 10 3 squared , middle dot , 3 super begin box , , end box end super , equals , 3 to the tenth
28. a 6 · a   = a 8 eh to the sixth , middle dot , eh super begin box , , end box end super , equals , eh to the eighth
29. x 2 y 5 · x   y   = x 5 y 11 x squared , y to the fifth , middle dot , x super begin box , , end box end super . y super begin box , , end box end super , equals , x to the fifth , y to the eleventh
30. ( 5 5 )   = 5 15 open , 5 to the fifth , close super begin box , , end box end super . equals , 5 to the fifteenth
31. ( b − 4 )   = b 20 open , b super negative 4 end super , close super begin box , , end box end super . equals , b to the twentieth
32. ( 4 x 3 y 5 )   = 16 x 6 y 10 open . 4 , x cubed , y to the fifth . close super begin box , , end box end super . equals 16 , x to the sixth , y to the tenth

Simplify each expression.

33. 2 d 2 · d 3 2 , d squared , middle dot , d cubed
34. ( q 3 r ) 4 open , q cubed , r , close to the fourth
35. ( 5 c − 4 ) ( − 4 m 2 c 8 ) open 5 , c super negative 4 end super , close open negative 4 , m squared , c to the eighth , close
36. ( 1.34 2 ) 5 ( 1.34 ) − 8 open , 1.34 squared , close to the fifth . open , 1.34 , close super negative 8 end super
37. ( 12 x 2 y − 2 ) 5 ( 4 x y − 3 ) − 7 open . 12 , x squared . y super negative 2 end super . close to the fifth . open . 4 x , y super negative 3 end super . close super negative 7 end super
38. ( − 2 r − 4 ) 2 ( − 3 r 2 z 8 ) − 1 open . negative 2 , r super negative 4 end super . close squared . open . negative 3 , r squared , z to the eighth . close super negative 1 end super
39. Estimation Each square inch of your body has about 6.5 × 10 2 6.5 times , 10 squared  pores. Suppose the back of your hand has an area of about 0.12 × 10 2 in . 2 . 0.12 , times , 10 squared , in , . squared , .  About how many pores are on the back of your hand? Write your answer in scientific notation.

7-5 Division Properties of Exponents

Quick Review

To divide powers with the same base, subtract the exponents.

a m a n = a m − n , fraction eh to the m , over eh to the n end fraction . equals . eh super m minus n end super . comma  where a ≠ 0 eh not equal to 0  and m and n are integers

To raise a quotient to a power, raise the numerator and the denominator to the power.

( a b ) n = a n b n , open , eh over b , close to the n . equals . fraction eh to the n , over b to the n end fraction . comma  where a ≠ 0 , eh not equal to 0 comma   b ≠ 0 , b not equal to 0 comma  and n is an integer

Example

What is the simplified form of ( 5 x 4 z 2 ) 3 ? open . fraction 5 , x to the fourth , over z squared end fraction . close cubed . question mark

( 5 x 4 z 2 ) = ( 5 x 4 ) 3 ( z 2 ) 3 = 5 3 x 4 ⋅ 3 z 2 ⋅ 3 = 125 x 12 z 6 open . fraction 5 , x to the fourth , over z squared end fraction . close . equals . fraction open , 5 , x to the fourth , close cubed , over open , z squared , close cubed end fraction . equals . fraction 5 cubed . x super 4 dot 3 end super , over z super 2 dot 3 end super end fraction . equals . fraction 125 , x to the twelfth , over z to the sixth end fraction

Exercises

Simplify each expression.

40. w 2 w 5 fraction w squared , over w to the fifth end fraction
41. 21 x 3 3 x − 1 fraction 21 , x cubed , over 3 , x super negative 1 end super end fraction
42. ( n 5 v 3 ) 7 open . fraction n to the fifth , over v cubed end fraction . close to the seventh
43. ( 3 c 3 e 5 ) − 4 open . fraction 3 , c cubed , over e to the fifth end fraction . close super negative 4 end super

Simplify each quotient. Write your answer in scientific notation.

44. 4 .2 × 10 8 2.1 × 10 11 fraction 4 .2 times , 10 to the eighth , over 2.1 times , 10 to the eleventh end fraction
45. 3.1 × 10 4 1.24 × 10 2 fraction 3.1 times , 10 to the fourth , over 1.24 , times , 10 squared end fraction
46. 4.5 × 10 3 9 × 10 7 fraction 4.5 times , 10 cubed , over 9 times , 10 to the seventh end fraction
47. 5.1 × 10 5 1.7 × 10 2 fraction 5.1 times , 10 to the fifth , over 1.7 times , 10 squared end fraction
48. Writing List the steps that you would use to simplify ( 5 a 8 10 a 6 ) − 3 . open . fraction 5 , eh to the eighth , over 10 , eh to the sixth end fraction . close super negative 3 end super . .

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