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Basic Properties of Real Numbers: The Power Rules for Exponents

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

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Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret. Objectives of this module: understand the power rules for powers, products, and quotients.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Overview

  • The Power Rule for Powers
  • The Power Rule for Products
  • The Power Rule for quotients

The Power Rule for Powers

The following examples suggest a rule for raising a power to a power:

Example 1

( a 2 ) 3 = a 2  ⋅ ​ a 2  ⋅  a 2 ( a 2 ) 3 = a 2  ⋅ ​ a 2  ⋅  a 2

Using the product rule we get

( a 2 ) 3 = a 2+2+2 ( a 2 ) 3 = a 3 ⋅ 2 ( a 2 ) 3 = a 6 ( a 2 ) 3 = a 2+2+2 ( a 2 ) 3 = a 3 ⋅ 2 ( a 2 ) 3 = a 6

Example 2

( x 9 ) 4 = x 9 ⋅ x 9 ⋅ x 9 ⋅ x 9 ( x 9 ) 4 = x 9+9+9+9 ( x 9 ) 4 = x 4 ⋅ 9 ( x 9 ) 4 = x 36 ( x 9 ) 4 = x 9 ⋅ x 9 ⋅ x 9 ⋅ x 9 ( x 9 ) 4 = x 9+9+9+9 ( x 9 ) 4 = x 4 ⋅ 9 ( x 9 ) 4 = x 36

POWER RULE FOR POWERS

If x x is a real number and n n and m m are natural numbers,
 ( x n ) m = x n⋅m ( x n ) m = x n⋅m

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

Sample Set A

Simplify each expression using the power rule for powers. All exponents are natural numbers.

Example 3

( x 3 ) 4 = x 3 ⋅ 4   x 12 The box represents a step done mentally. ( x 3 ) 4 = x 3 ⋅ 4   x 12 The box represents a step done mentally.

Example 4

( y 5 ) 3 = y 5 ⋅ 3 = y 15 ( y 5 ) 3 = y 5 ⋅ 3 = y 15

Example 5

( d 20 ) 6 = d 20 ⋅ 6 = d 120 ( d 20 ) 6 = d 20 ⋅ 6 = d 120

Example 6

( x □ ) △ =  x □△ ( x □ ) △ =  x □△

Although we don’t know exactly what number □△ □△ is, the notation □△ □△ indicates the multiplication.

Practice Set A

Simplify each expression using the power rule for powers.

Exercise 1

( x 5 ) 4 ( x 5 ) 4

Solution

x 20 x 20

Exercise 2

( y 7 ) 7 ( y 7 ) 7

Solution

y 49 y 49

The Power Rule for Products

The following examples suggest a rule for raising a product to a power:

Example 7

(ab) 3 = ab⋅ab⋅ab Use ​the commutative property of multiplication. = aaabbb = a 3 b 3 (ab) 3 = ab⋅ab⋅ab Use ​the commutative property of multiplication. = aaabbb = a 3 b 3

Example 8

(xy) 5 = xy⋅xy⋅xy⋅xy⋅xy = xxxxx⋅yyyyy = x 5 y 5 (xy) 5 = xy⋅xy⋅xy⋅xy⋅xy = xxxxx⋅yyyyy = x 5 y 5

Example 9

(4x y z ) 2 = 4xyz⋅4xyz = 4⋅4⋅xx⋅yy⋅zz = 16 x 2 y 2 z 2 (4x y z ) 2 = 4xyz⋅4xyz = 4⋅4⋅xx⋅yy⋅zz = 16 x 2 y 2 z 2

POWER RULE FOR PRODUCTS

If x x and y y are real numbers and n n is a natural number,
 (xy) n = x n y n (xy) n = x n y n

To raise a product to a power, apply the exponent to each and every factor.

Sample Set B

Make use of either or both the power rule for products and power rule for powers to simplify each expression.

Example 10

(ab) 7 = a 7 b 7 (ab) 7 = a 7 b 7

Example 11

(axy) 4 = a 4 x 4 y 4 (axy) 4 = a 4 x 4 y 4

Example 12

(3ab) 2 = 3 2 a 2 b 2 =9 a 2 b 2 Don't forget to apply the exponent to the 3! (3ab) 2 = 3 2 a 2 b 2 =9 a 2 b 2 Don't forget to apply the exponent to the 3!

Example 13

(2st) 5 = 2 5 s 5 t 5 = 32 s 5 t 5 (2st) 5 = 2 5 s 5 t 5 = 32 s 5 t 5

Example 14

(a b 3 ) 2 = a 2 ( b 3 ) 2 = a 2 b 6 We used two rules here. First, the power rule for products. Second, the power rule for powers. (a b 3 ) 2 = a 2 ( b 3 ) 2 = a 2 b 6 We used two rules here. First, the power rule for products. Second, the power rule for powers.

Example 15

(7 a 4 b 2 c 8 ) 2 = 7 2 ( a 4 ) 2 ( b 2 ) 2 ( c 8 ) 2 =49 a 8 b 4 c 16 (7 a 4 b 2 c 8 ) 2 = 7 2 ( a 4 ) 2 ( b 2 ) 2 ( c 8 ) 2 =49 a 8 b 4 c 16

Example 16

If 6 a 3 c 7 ≠0, then  (6 a 3 c 7 ) 0 =1 Recall that  x 0 =1 for x≠0. If 6 a 3 c 7 ≠0, then  (6 a 3 c 7 ) 0 =1 Recall that  x 0 =1 for x≠0.

Example 17

[ 2 ( x+1 ) 4 ] 6 = 2 6 (x+1) 24 =64 (x+1) 24 [ 2 ( x+1 ) 4 ] 6 = 2 6 (x+1) 24 =64 (x+1) 24

Practice Set B

Make use of either or both the power rule for products and the power rule for powers to simplify each expression.

Exercise 3

(ax) 4 (ax) 4

Solution

a 4 x 4 a 4 x 4

Exercise 4

(3bxy) 2 (3bxy) 2

Solution

9 b 2 x 2 y 2 9 b 2 x 2 y 2

Exercise 5

[ 4t( s−5 ) ] 3 [ 4t( s−5 ) ] 3

Solution

64 t 3 ( s−5 ) 3 64 t 3 ( s−5 ) 3

Exercise 6

(9 x 3 y 5 ) 2 (9 x 3 y 5 ) 2

Solution

81 x 6 y 10 81 x 6 y 10

Exercise 7

(1 a 5 b 8 c 3 d) 6 (1 a 5 b 8 c 3 d) 6

Solution

a 30 b 48 c 18 d 6 a 30 b 48 c 18 d 6

Exercise 8

[ ( a+8 )( a+5 ) ] 4 [ ( a+8 )( a+5 ) ] 4

Solution

( a+8 ) 4 ( a+5 ) 4 ( a+8 ) 4 ( a+5 ) 4

Exercise 9

[ (12 c 4 u 3 (w−3) 2 ] 5 [ (12 c 4 u 3 (w−3) 2 ] 5

Solution

12 5 c 20 u 15 (w−3) 10 12 5 c 20 u 15 (w−3) 10

Exercise 10

[ 10 t 4 y 7 j 3 d 2 v 6 n 4 g 8 ( 2−k ) 17 ] 4 [ 10 t 4 y 7 j 3 d 2 v 6 n 4 g 8 ( 2−k ) 17 ] 4

Solution

10 4 t 16 y 28 j 12 d 8 v 24 n 16 g 32 (2−k) 68 10 4 t 16 y 28 j 12 d 8 v 24 n 16 g 32 (2−k) 68

Exercise 11

( x 3 x 5 y 2 y 6 ) 9 ( x 3 x 5 y 2 y 6 ) 9

Solution

( x 8 y 8 ) 9 = x 72 y 72 ( x 8 y 8 ) 9 = x 72 y 72

Exercise 12

( 10 6 ⋅ 10 12 ⋅ 10 5 ) 10 ( 10 6 ⋅ 10 12 ⋅ 10 5 ) 10

Solution

10 230 10 230

The Power Rule for Quotients

The following example suggests a rule for raising a quotient to a power.

Example 18

( a b ) 3 = a b ⋅ a b ⋅ a b = a⋅a⋅a b⋅b⋅b = a 3 b 3 ( a b ) 3 = a b ⋅ a b ⋅ a b = a⋅a⋅a b⋅b⋅b = a 3 b 3

POWER RULE FOR QUOTIENTS

If x x and y y are real numbers and n n is a natural number,
 ( x y ) n = x n y n ,  y≠0 ( x y ) n = x n y n ,  y≠0

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

Sample Set C

Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression. All exponents are natural numbers.

Example 19

( x y ) 6 = x 6 y 6 ( x y ) 6 = x 6 y 6

Example 20

( a c ) 2 = a 2 c 2 ( a c ) 2 = a 2 c 2

Example 21

( 2x b ) 4 = ( 2x ) 4 b 4 = 2 4 x 4 b 4 = 16 x 4 b 4 ( 2x b ) 4 = ( 2x ) 4 b 4 = 2 4 x 4 b 4 = 16 x 4 b 4

Example 22

( a 3 b 5 ) 7 = ( a 3 ) 7 ( b 5 ) 7 = a 21 b 35 ( a 3 b 5 ) 7 = ( a 3 ) 7 ( b 5 ) 7 = a 21 b 35

Example 23

( 3 c 4 r 2 2 3 g 5 ) 3 = 3 3 c 12 r 6 2 9 g 15 = 27 c 12 r 6 2 9 g 15 or 27 c 12 r 6 512 g 15 ( 3 c 4 r 2 2 3 g 5 ) 3 = 3 3 c 12 r 6 2 9 g 15 = 27 c 12 r 6 2 9 g 15 or 27 c 12 r 6 512 g 15

Example 24

[ ( a−2 ) ( a+7 ) ] 4 = ( a−2 ) 4 ( a+7 ) 4 [ ( a−2 ) ( a+7 ) ] 4 = ( a−2 ) 4 ( a+7 ) 4

Example 25

[ 6x ( 4−x ) 4 2a ( y−4 ) 6 ] 2 = 6 2 x 2 ( 4−x ) 8 2 2 a 2 ( y−4 ) 12 = 36 x 2 ( 4−x ) 8 4 a 2 ( y−4 ) 12 = 9 x 2 ( 4−x ) 8 a 2 ( y−4 ) 12 [ 6x ( 4−x ) 4 2a ( y−4 ) 6 ] 2 = 6 2 x 2 ( 4−x ) 8 2 2 a 2 ( y−4 ) 12 = 36 x 2 ( 4−x ) 8 4 a 2 ( y−4 ) 12 = 9 x 2 ( 4−x ) 8 a 2 ( y−4 ) 12

Example 26

( a 3 b 5 a 2 b ) 3 = ( a 3−2 b 5−1 ) 3 We can simplify within the parentheses. We have a rule that tells us to proceed this way. = (a b 4 ) 3 = a 3 b 12 ( a 3 b 5 a 2 b ) 3 = a 9 b 15 a 6 b 3 = a 9−6 b 15−3 = a 3 b 12 We could have actually used the power rule for quotients first. Distribute the exponent, then simplify using the other rules. It is probably better, for the sake of consistency, to work inside the parentheses first. ( a 3 b 5 a 2 b ) 3 = ( a 3−2 b 5−1 ) 3 We can simplify within the parentheses. We have a rule that tells us to proceed this way. = (a b 4 ) 3 = a 3 b 12 ( a 3 b 5 a 2 b ) 3 = a 9 b 15 a 6 b 3 = a 9−6 b 15−3 = a 3 b 12 We could have actually used the power rule for quotients first. Distribute the exponent, then simplify using the other rules. It is probably better, for the sake of consistency, to work inside the parentheses first.

Example 27

( a r b s c t ) w = a rw b sw c tw   ( a r b s c t ) w = a rw b sw c tw  

Practice Set C

Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression.

Exercise 13

( a c ) 5 ( a c ) 5

Solution

a 5 c 5 a 5 c 5

Exercise 14

( 2x 3y ) 3 ( 2x 3y ) 3

Solution

8 x 3 27 y 3 8 x 3 27 y 3

Exercise 15

( x 2 y 4 z 7 a 5 b ) 9 ( x 2 y 4 z 7 a 5 b ) 9

Solution

x 18 y 36 z 63 a 45 b 9 x 18 y 36 z 63 a 45 b 9

Exercise 16

[ 2 a 4 ( b−1 ) 3 b 3 ( c+6 ) ] 4 [ 2 a 4 ( b−1 ) 3 b 3 ( c+6 ) ] 4

Solution

16 a 16 ( b−1 ) 4 81 b 12 ( c+6 ) 4 16 a 16 ( b−1 ) 4 81 b 12 ( c+6 ) 4

Exercise 17

( 8 a 3 b 2 c 6 4 a 2 b ) 3 ( 8 a 3 b 2 c 6 4 a 2 b ) 3

Solution

8 a 3 b 3 c 18 8 a 3 b 3 c 18

Exercise 18

[ ( 9+w ) 2 ( 3+w ) 5 ] 10 [ ( 9+w ) 2 ( 3+w ) 5 ] 10

Solution

( 9+w ) 20 ( 3+w ) 50 ( 9+w ) 20 ( 3+w ) 50

Exercise 19

[ 5 x 4 ( y+1 ) 5 x 4 ( y+1 ) ] 6 [ 5 x 4 ( y+1 ) 5 x 4 ( y+1 ) ] 6

Solution

1, if  x 4 (y+1)≠0 1, if  x 4 (y+1)≠0

Exercise 20

( 16 x 3 v 4 c 7 12 x 2 v c 6 ) 0 ( 16 x 3 v 4 c 7 12 x 2 v c 6 ) 0

Solution

1, if  x 2 v c 6 ≠0 1, if  x 2 v c 6 ≠0

Exercises

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers.

Exercise 21

( ac ) 5 ( ac ) 5

Solution

a 5 c 5 a 5 c 5

Exercise 22

( nm ) 7 ( nm ) 7

Exercise 23

( 2a ) 3 ( 2a ) 3

Solution

8 a 3 8 a 3

Exercise 24

( 2a ) 5 ( 2a ) 5

Exercise 25

( 3xy ) 4 ( 3xy ) 4

Solution

81 x 4 y 4 81 x 4 y 4

Exercise 26

( 2xy ) 5 ( 2xy ) 5

Exercise 27

( 3ab ) 4 ( 3ab ) 4

Solution

81 a 4 b 4 81 a 4 b 4

Exercise 28

( 6mn ) 2 ( 6mn ) 2

Exercise 29

(7 y 3 ) 2 (7 y 3 ) 2

Solution

49 y 6 49 y 6

Exercise 30

(3 m 3 ) 4 (3 m 3 ) 4

Exercise 31

(5 x 6 ) 3 (5 x 6 ) 3

Solution

125 x 18 125 x 18

Exercise 32

(5 x 2 ) 3 (5 x 2 ) 3

Exercise 33

(10 a 2 b) 2 (10 a 2 b) 2

Solution

100 a 4 b 2 100 a 4 b 2

Exercise 34

(8 x 2 y 3 ) 2 (8 x 2 y 3 ) 2

Exercise 35

( x 2 y 3 z 5 ) 4 ( x 2 y 3 z 5 ) 4

Solution

x 8 y 12 z 20 x 8 y 12 z 20

Exercise 36

(2 a 5 b 11 ) 0 (2 a 5 b 11 ) 0

Exercise 37

( x 3 y 2 z 4 ) 5 ( x 3 y 2 z 4 ) 5

Solution

x 15 y 10 z 20 x 15 y 10 z 20

Exercise 38

( m 6 n 2 p 5 ) 5 ( m 6 n 2 p 5 ) 5

Exercise 39

( a 4 b 7 c 6 d 8 ) 8 ( a 4 b 7 c 6 d 8 ) 8

Solution

a 32 b 56 c 48 d 64 a 32 b 56 c 48 d 64

Exercise 40

( x 2 y 3 z 9 w 7 ) 3 ( x 2 y 3 z 9 w 7 ) 3

Exercise 41

(9x y 3 ) 0 (9x y 3 ) 0

Solution

1

Exercise 42

( 1 2 f 2 r 6 s 5 ) 4 ( 1 2 f 2 r 6 s 5 ) 4

Exercise 43

( 1 8 c 10 d 8 e 4 f 9 ) 2 ( 1 8 c 10 d 8 e 4 f 9 ) 2

Solution

1 64 c 20 d 16 e 8 f 18 1 64 c 20 d 16 e 8 f 18

Exercise 44

( 3 5 a 3 b 5 c 10 ) 3 ( 3 5 a 3 b 5 c 10 ) 3

Exercise 45

(xy) 4 ( x 2 y 4 ) (xy) 4 ( x 2 y 4 )

Solution

x 6 y 8 x 6 y 8

Exercise 46

(2 a 2 ) 4 (3 a 5 ) 2 (2 a 2 ) 4 (3 a 5 ) 2

Exercise 47

( a 2 b 3 ) 3 ( a 3 b 3 ) 4 ( a 2 b 3 ) 3 ( a 3 b 3 ) 4

Solution

a 18 b 21 a 18 b 21

Exercise 48

( h 3 k 5 ) 2 ( h 2 k 4 ) 3 ( h 3 k 5 ) 2 ( h 2 k 4 ) 3

Exercise 49

( x 4 y 3 z) 4 ( x 5 y z 2 ) 2 ( x 4 y 3 z) 4 ( x 5 y z 2 ) 2

Solution

x 26 y 14 z 8 x 26 y 14 z 8

Exercise 50

(a b 3 c 2 ) 5 ( a 2 b 2 c) 2 (a b 3 c 2 ) 5 ( a 2 b 2 c) 2

Exercise 51

(6 a 2 b 8 ) 2 (3a b 5 ) 2 (6 a 2 b 8 ) 2 (3a b 5 ) 2

Solution

4 a 2 b 6 4 a 2 b 6

Exercise 52

( a 3 b 4 ) 5 ( a 4 b 4 ) 3 ( a 3 b 4 ) 5 ( a 4 b 4 ) 3

Exercise 53

( x 6 y 5 ) 3 ( x 2 y 3 ) 5 ( x 6 y 5 ) 3 ( x 2 y 3 ) 5

Solution

x 8 x 8

Exercise 54

( a 8 b 10 ) 3 ( a 7 b 5 ) 3 ( a 8 b 10 ) 3 ( a 7 b 5 ) 3

Exercise 55

( m 5 n 6 p 4 ) 4 ( m 4 n 5 p ) 4 ( m 5 n 6 p 4 ) 4 ( m 4 n 5 p ) 4

Solution

m 4 n 4 p 12 m 4 n 4 p 12

Exercise 56

( x 8 y 3 z 2 ) 5 ( x 6 yz ) 6 ( x 8 y 3 z 2 ) 5 ( x 6 yz ) 6

Exercise 57

( 10 x 4 y 5 z 11 ) 3 ( x y 2 ) 4 ( 10 x 4 y 5 z 11 ) 3 ( x y 2 ) 4

Solution

1000 x 8 y 7 z 33 1000 x 8 y 7 z 33

Exercise 58

(9 a 4 b 5 )(2 b 2 c) (3 a 3 b)(6bc) (9 a 4 b 5 )(2 b 2 c) (3 a 3 b)(6bc)

Exercise 59

( 2 x 3 y 3 ) 4 ( 5 x 6 y 8 ) 2 ( 4 x 5 y 3 ) 2 ( 2 x 3 y 3 ) 4 ( 5 x 6 y 8 ) 2 ( 4 x 5 y 3 ) 2

Solution

25 x 14 y 22 25 x 14 y 22

Exercise 60

( 3x 5y ) 2 ( 3x 5y ) 2

Exercise 61

( 3ab 4xy ) 3 ( 3ab 4xy ) 3

Solution

27 a 3 b 3 64 x 3 y 3 27 a 3 b 3 64 x 3 y 3

Exercise 62

( x 2 y 2 2 z 3 ) 5 ( x 2 y 2 2 z 3 ) 5

Exercise 63

( 3 a 2 b 3 c 4 ) 3 ( 3 a 2 b 3 c 4 ) 3

Solution

27 a 6 b 9 c 12 27 a 6 b 9 c 12

Exercise 64

( 4 2 a 3 b 7 b 5 c 4 ) 2 ( 4 2 a 3 b 7 b 5 c 4 ) 2

Exercise 65

[ x 2 ( y−1 ) 3 ( x+6 ) ] 4 [ x 2 ( y−1 ) 3 ( x+6 ) ] 4

Solution

x 8 ( y−1 ) 12 ( x+6 ) 4 x 8 ( y−1 ) 12 ( x+6 ) 4

Exercise 66

( x n t 2m ) 4 ( x n t 2m ) 4

Exercise 67

( x n+2 ) 3 x 2n ( x n+2 ) 3 x 2n

Solution

x n+6 x n+6

Exercise 68

( xy ) △ ( xy ) △

Exercise 70

'Three a to the power triangle b to the power delta, the whole to the power square' over 'five x y to the power rhombus' the whole to the power star.'

Exercise 72

4 3 a Δ a □ 4 a ∇ 4 3 a Δ a □ 4 a ∇

Exercise 73

( 4 x Δ 2 y ∇ ) □ ( 4 x Δ 2 y ∇ ) □

Solution

'Two to the power square, x to the power the product of triangle and square' over 'y to the power the product of delta and square'.

Exercise 74

'Sixteen a cube b to the power star' over 'five a to the power triangle b to the power delta'. The whole to the zeroth power.

Exercises for Review

Exercise 75

((Reference)) Is there a smallest integer? If so, what is it?

Solution

no

Exercise 76

((Reference)) Use the distributive property to expand 5a( 2x+8 ) 5a( 2x+8 ) .

Exercise 77

((Reference)) Find the value of ( 5−3 ) 2 + ( 5+4 ) 3 +2 4 2 −2⋅5−1 ( 5−3 ) 2 + ( 5+4 ) 3 +2 4 2 −2⋅5−1 .

Solution

147

Exercise 78

((Reference)) Assuming the bases are not zero, find the value of (4 a 2 b 3 )(5a b 4 ) (4 a 2 b 3 )(5a b 4 ) .

Exercise 79

((Reference)) Assuming the bases are not zero, find the value of 36 x 10 y 8 z 3 w 0 9 x 5 y 2 z 36 x 10 y 8 z 3 w 0 9 x 5 y 2 z .

Solution

4 x 5 y 6 z 2 4 x 5 y 6 z 2

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