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Basic Properties of Real Numbers: Rules of Exponents

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

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 product and quotient rules for exponents, understand the meaning of zero as an exponent.

Overview

  • The Product Rule for Exponents
  • The Quotient Rule for Exponents
  • Zero as an Exponent

We will begin our study of the rules of exponents by recalling the definition of exponents.

Definition of Exponents

If x x is any real number and n n is a natural number, then

 x n = x⋅x⋅x⋅...⋅x ︸ n factors of x  x n = x⋅x⋅x⋅...⋅x ︸ n factors of x 

An exponent records the number of identical factors in a multiplication.

Base Exponent Power

In x n x n ,

 x x is the base
 n n is the exponent
The number represented by x n x n is called a power.

The term x n x n is read as " x x to the n n th."

The Product Rule for Exponents

The first rule we wish to develop is the rule for multiplying two exponential quantities having the same base and natural number exponents. The following examples suggest this rule:

Example 1

x 2 ⋅ x 4 = xx ︸ ⋅ xxxx ︸ = xxxxxx ︸ = x 6  2 + 4 = 6                                              factors        factors x 2 ⋅ x 4 = xx ︸ ⋅ xxxx ︸ = xxxxxx ︸ = x 6  2 + 4 = 6                                              factors        factors

Example 2

a ⋅ a 2 = a ︸ ⋅ aa ︸ = aaa ︸ = a 3 1 + 2 = 3                                      factors   factors a ⋅ a 2 = a ︸ ⋅ aa ︸ = aaa ︸ = a 3 1 + 2 = 3                                      factors   factors

PRODUCT RULE FOR EXPONENTS

If x x is a real number and n n and m m are natural numbers,

 x n x m = x n+m x n x m = x n+m

To multiply two exponential quantities having the same base, add the exponents. Keep in mind that the exponential quantities being multiplied must have the same base for this rule to apply.

Sample Set A

Find the following products. All exponents are natural numbers.

Example 3

x 3 ⋅ x 5 = x 3+5 = x 8 x 3 ⋅ x 5 = x 3+5 = x 8

Example 4

a 6 ⋅ a 14 = a 6+14 = a 20 a 6 ⋅ a 14 = a 6+14 = a 20

Example 5

y 5 ⋅y= y 5 ⋅ y 1 = y 5+1 = y 6 y 5 ⋅y= y 5 ⋅ y 1 = y 5+1 = y 6

Example 6

(x−2y) 8 (x−2y) 5 = (x−2y) 8+5 = (x−2y) 13 (x−2y) 8 (x−2y) 5 = (x−2y) 8+5 = (x−2y) 13

Example 7

x 3 y 4 ≠ (xy) 3+4 Since the bases are not the same, the  product rule does not apply. x 3 y 4 ≠ (xy) 3+4 Since the bases are not the same, the  product rule does not apply.

Practice Set A

Find each product.

Exercise 1

x 2 ⋅ x 5 x 2 ⋅ x 5

Solution

x 2+5 = x 7 x 2+5 = x 7

Exercise 2

x 9 ⋅ x 4 x 9 ⋅ x 4

Solution

x 9+4 = x 13 x 9+4 = x 13

Exercise 3

y 6 ⋅ y 4 y 6 ⋅ y 4

Solution

y 6+4 = y 10 y 6+4 = y 10

Exercise 4

c 12 ⋅ c 8 c 12 ⋅ c 8

Solution

c 12+8 = c 20 c 12+8 = c 20

Exercise 5

(x+2) 3 ⋅ (x+2) 5 (x+2) 3 ⋅ (x+2) 5

Solution

(x+2) 3+5 = (x+2) 8 (x+2) 3+5 = (x+2) 8

Sample Set B

We can use the first rule of exponents (and the others that we will develop) along with the properties of real numbers.

Example 8

2 x 3 ⋅7 x 5 = 2⋅7⋅ x 3+5 =14 x 8 2 x 3 ⋅7 x 5 = 2⋅7⋅ x 3+5 =14 x 8

We used the commutative and associative properties of multiplication. In practice, we use these properties “mentally” (as signified by the drawing of the box). We don’t actually write the second step.

Example 9

4 y 3 ⋅6 y 2 = 4⋅6⋅ y 3+2 =24 y 5 4 y 3 ⋅6 y 2 = 4⋅6⋅ y 3+2 =24 y 5

Example 10

9 a 2 b 6 (8a b 4 2 b 3 )= 9⋅8⋅2 a 2+1 b 6+4+3 =144 a 3 b 13 9 a 2 b 6 (8a b 4 2 b 3 )= 9⋅8⋅2 a 2+1 b 6+4+3 =144 a 3 b 13

Example 11

5 (a+6) 2 ⋅3 (a+6) 8 = 5⋅3 (a+6) 2+8 =15 (a+6) 10 5 (a+6) 2 ⋅3 (a+6) 8 = 5⋅3 (a+6) 2+8 =15 (a+6) 10

Example 12

4 x 3 ⋅12⋅ y 2 =48 x 3 y 2 4 x 3 ⋅12⋅ y 2 =48 x 3 y 2

Example 13

The product of four a to the power triangle, and five a to the power star is equal to twenty a to the power 'triangle plus star'.

The bases are the same, so we add the exponents. Although we don’t know exactly what number Sum of a triangle and a star. is, the notation Sum of a triangle and a star. indicates the addition.

Practice Set B

Perform each multiplication in one step.

Exercise 6

3 x 5 ⋅2 x 2 3 x 5 ⋅2 x 2

Solution

6 x 7 6 x 7

Exercise 7

6 y 3 ⋅3 y 4 6 y 3 ⋅3 y 4

Solution

18 y 7 18 y 7

Exercise 8

4 a 3 b 2 ⋅9 a 2 b 4 a 3 b 2 ⋅9 a 2 b

Solution

36 a 5 b 3 36 a 5 b 3

Exercise 9

x 4 ⋅4 y 2 ⋅2 x 2 ⋅7 y 6 x 4 ⋅4 y 2 ⋅2 x 2 ⋅7 y 6

Solution

56 x 6 y 8 56 x 6 y 8

Exercise 10

(x−y) 3 ⋅4 (x−y) 2 (x−y) 3 ⋅4 (x−y) 2

Solution

4 (x−y) 5 4 (x−y) 5

Exercise 11

8 x 4 y 2 x x 3 y 5 8 x 4 y 2 x x 3 y 5

Solution

8 x 8 y 7 8 x 8 y 7

Exercise 12

2aa a 3 (a b 2 a 3 )b6a b 2 2aa a 3 (a b 2 a 3 )b6a b 2

Solution

12 a 10 b 5 12 a 10 b 5

Exercise 13

a n ⋅ a m ⋅ a r a n ⋅ a m ⋅ a r

Solution

a n+m+r a n+m+r

The Quotient Rule for Exponents

The second rule we wish to develop is the rule for dividing two exponential quantities having the same base and natural number exponents.
 The following examples suggest a rule for dividing two exponential quantities having the same base and natural number exponents.

Example 14

x 5 x 2 = xxxxx xx = ( xx )xxx ( xx ) =xxx= x 3 . Notice that 5−2=3. x 5 x 2 = xxxxx xx = ( xx )xxx ( xx ) =xxx= x 3 . Notice that 5−2=3.

Example 15

a 8 a 3 = aaaaaaaa aaa = ( aaa )aaaaa ( aaa ) =aaaaa= a 5 . Notice that 8−3=5. a 8 a 3 = aaaaaaaa aaa = ( aaa )aaaaa ( aaa ) =aaaaa= a 5 . Notice that 8−3=5.

QUOTIENT RULE FOR EXPONENTS

If x x is a real number and n n and m m are natural numbers,

 x n x m = x n−m , x≠0 x n x m = x n−m , x≠0 .

To divide two exponential quantities having the same nonzero base, subtract the exponent of the denominator from the exponent of the numerator. Keep in mind that the exponential quantities being divided must have the same base for this rule to apply.

Sample Set C

Find the following quotients. All exponents are natural numbers.

Example 16

x 5 x 2 = x 5-2 = x 3 The part in the box is usally done mentally. x 5 x 2 = x 5-2 = x 3 The part in the box is usally done mentally.

Example 17

27 a 3 b 6 c 2 3 a 2 bc = 9 a 3−2 b 6−1 c 2−1 =9a b 5 c 27 a 3 b 6 c 2 3 a 2 bc = 9 a 3−2 b 6−1 c 2−1 =9a b 5 c

Example 18

15 x □ 3 x △ =5 x □−△ 15 x □ 3 x △ =5 x □−△

The bases are the same, so we subtract the exponents. Although we don’t know exactly what □−△ □−△ is, the notation □−△ □−△ indicates the subtraction.

Practice Set C

Find each quotient

Exercise 14

y 9 y 5 y 9 y 5

Solution

y 4 y 4

Exercise 15

a 7 a a 7 a

Solution

a 6 a 6

Exercise 16

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

Solution

(x+6) 2 (x+6) 2

Exercise 17

26 x 4 y 6 z 2 13 x 2 y 2 z 26 x 4 y 6 z 2 13 x 2 y 2 z

Solution

2 x 2 y 4 z 2 x 2 y 4 z

When we make the subtraction, n−m n−m , in the division x n x m x n x m , there are three possibilities for the values of the exponents:

  1. The exponent of the numerator is greater than the exponent of the denominator, that is, n>m n>m . Thus, the exponent, n−m n−m , is a natural number.
  2. The exponents are the same, that is, n=m n=m . Thus, the exponent, n−m n−m , is zero, a whole number.
  3. The exponent of the denominator is greater than the exponent of the numerator, that is, n<m n<m . Thus, the exponent, n−m n−m , is an integer.

Zero as an Exponent

In Sample Set C, the exponents of the numerators were greater than the exponents of the denominators. Let’s study the case when the exponents are the same.

When the exponents are the same, say n n , the subtraction n−n n−n produces 0.

Thus, by the second rule of exponents, x n x n = x n−n = x 0 x n x n = x n−n = x 0 .

But what real number, if any, does x 0 x 0 represent? Let’s think for a moment about our experience with division in arithmetic. We know that any nonzero number divided by itself is one.

8 8 =1, 43 43 =1, 258 258 =1 8 8 =1, 43 43 =1, 258 258 =1

Since the letter x x represents some nonzero real number, so does x n x n . Thus, x n x n x n x n

represents some nonzero real number divided by itself. Then x n x n =1 x n x n =1 .

But we have also established that if x≠0,  x n x n = x 0 x≠0,  x n x n = x 0 . We now have that   x n x n = x 0   x n x n = x 0

and   x n x n =1   x n x n =1 . This implies that x 0 =1,x≠0 x 0 =1,x≠0 .

Exponents can now be natural numbers and zero. We have enlarged our collection of numbers that can be used as exponents from the collection of natural numbers to the collection of whole numbers.

ZERO AS AN EXPONENT

If x≠0, x 0 =1 x≠0, x 0 =1

Any number, other than 0, raised to the power of 0, is 1. 0 0 0 0 has no meaning (it does not represent a number).

Sample Set D

Find each value. Assume the base is not zero.

Example 19

6 0 =1 6 0 =1

Example 20

247 0 =1 247 0 =1

Example 21

(2a+5) 0 =1 (2a+5) 0 =1

Example 22

4 y 0 =4⋅1=4 4 y 0 =4⋅1=4

Example 23

y 6 y 6 = y 0 =1 y 6 y 6 = y 0 =1

Example 24

2 x 2 x 2 =2 x 0 =2⋅1=2 2 x 2 x 2 =2 x 0 =2⋅1=2

Example 25

5 (x+4) 8 (x−1) 5 5 (x+4) 3 (x−1) 5 = (x+4) 8−3 (x−1) 5−5 = (x+4) 5 (x−1) 0 = (x+4) 5 5 (x+4) 8 (x−1) 5 5 (x+4) 3 (x−1) 5 = (x+4) 8−3 (x−1) 5−5 = (x+4) 5 (x−1) 0 = (x+4) 5

Practice Set D

Find each value. Assume the base is not zero.

Exercise 18

y 7 y 3 y 7 y 3

Solution

y 7−3 = y 4 y 7−3 = y 4

Exercise 19

6 x 4 2 x 3 6 x 4 2 x 3

Solution

3 x 4−3 =3x 3 x 4−3 =3x

Exercise 20

14 a 7 7 a 2 14 a 7 7 a 2

Solution

2 a 7−2 =2 a 5 2 a 7−2 =2 a 5

Exercise 21

26 x 2 y 5 4x y 2 26 x 2 y 5 4x y 2

Solution

13 2 x y 3 13 2 x y 3

Exercise 22

36 a 4 b 3 c 8 8a b 3 c 6 36 a 4 b 3 c 8 8a b 3 c 6

Solution

9 2 a 3 c 2 9 2 a 3 c 2

Exercise 23

51 (a−4) 3 17(a−4) 51 (a−4) 3 17(a−4)

Solution

3 (a−4) 2 3 (a−4) 2

Exercise 24

52 a 7 b 3 (a+b) 8 26 a 2 b (a+b) 8 52 a 7 b 3 (a+b) 8 26 a 2 b (a+b) 8

Solution

2 a 5 b 2 2 a 5 b 2

Exercise 25

a n a 3 a n a 3

Solution

a n−3 a n−3

Exercise 26

14 x r y p z q 2 x r y h z 5 14 x r y p z q 2 x r y h z 5

Solution

7 y p−h z q−5 7 y p−h z q−5

We will study the case where the exponent of the denominator is greater than the exponent of the numerator in Section (Reference).

Exercises

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers.

Exercise 27

3 2 ⋅ 3 3 3 2 ⋅ 3 3

Solution

3 5 =243 3 5 =243

Exercise 28

5 2 ⋅ 5 4 5 2 ⋅ 5 4

Exercise 29

9 0 ⋅ 9 2 9 0 ⋅ 9 2

Solution

9 2 =81 9 2 =81

Exercise 30

7 3 ⋅ 7 0 7 3 ⋅ 7 0

Exercise 31

2 4 ⋅ 2 5 2 4 ⋅ 2 5

Solution

2 9 =512 2 9 =512

Exercise 32

x 5 x 4 x 5 x 4

Exercise 33

x 2 x 3 x 2 x 3

Solution

x 5 x 5

Exercise 34

a 9 a 7 a 9 a 7

Exercise 35

y 5 y 7 y 5 y 7

Solution

y 12 y 12

Exercise 36

m 10 m 2 m 10 m 2

Exercise 37

k 8 k 3 k 8 k 3

Solution

k 11 k 11

Exercise 38

y 3 y 4 y 6 y 3 y 4 y 6

Exercise 39

3 x 2 ⋅2 x 5 3 x 2 ⋅2 x 5

Solution

6 x 7 6 x 7

Exercise 40

a 2 a 3 a 8 a 2 a 3 a 8

Exercise 41

4 y 4 ⋅5 y 6 4 y 4 ⋅5 y 6

Solution

20 y 10 20 y 10

Exercise 42

2 a 3 b 2 ⋅3ab 2 a 3 b 2 ⋅3ab

Exercise 43

12x y 3 z 2 ⋅4 x 2 y 2 z⋅3x 12x y 3 z 2 ⋅4 x 2 y 2 z⋅3x

Solution

144 x 4 y 5 z 3 144 x 4 y 5 z 3

Exercise 44

(3ab)(2 a 2 b) (3ab)(2 a 2 b)

Exercise 45

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

Solution

32 x 3 y 3 32 x 3 y 3

Exercise 46

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

Exercise 47

( 1 4 a 2 b 4 ) ( 1 2 b 4 ) ( 1 4 a 2 b 4 ) ( 1 2 b 4 )

Solution

1 8 a 2 b 8 1 8 a 2 b 8

Exercise 48

( 3 8 ) ( 16 21 x 2 y 3 ) ( x 3 y 2 ) ( 3 8 ) ( 16 21 x 2 y 3 ) ( x 3 y 2 )

Exercise 49

8 5 8 3 8 5 8 3

Solution

8 2 =64 8 2 =64

Exercise 50

6 4 6 3 6 4 6 3

Exercise 51

2 9 2 4 2 9 2 4

Solution

2 5 =32 2 5 =32

Exercise 52

4 16 4 13 4 16 4 13

Exercise 53

x 5 x 3 x 5 x 3

Solution

x 2 x 2

Exercise 54

y 4 y 3 y 4 y 3

Exercise 55

y 9 y 4 y 9 y 4

Solution

y 5 y 5

Exercise 56

k 16 k 13 k 16 k 13

Exercise 57

x 4 x 2 x 4 x 2

Solution

x 2 x 2

Exercise 58

y 5 y 2 y 5 y 2

Exercise 59

m 16 m 9 m 16 m 9

Solution

m 7 m 7

Exercise 60

a 9 b 6 a 5 b 2 a 9 b 6 a 5 b 2

Exercise 61

y 3 w 10 y w 5 y 3 w 10 y w 5

Solution

y 2 w 5 y 2 w 5

Exercise 62

m 17 n 12 m 16 n 10 m 17 n 12 m 16 n 10

Exercise 63

x 5 y 7 x 3 y 4 x 5 y 7 x 3 y 4

Solution

x 2 y 3 x 2 y 3

Exercise 64

15 x 20 y 24 z 4 5 x 19 yz 15 x 20 y 24 z 4 5 x 19 yz

Exercise 65

e 11 e 11 e 11 e 11

Solution

e 0 =1 e 0 =1

Exercise 66

6 r 4 6 r 4 6 r 4 6 r 4

Exercise 67

x 0 x 0 x 0 x 0

Solution

x 0 =1 x 0 =1

Exercise 68

a 0 b 0 c 0 a 0 b 0 c 0

Exercise 69

8 a 4 b 0 4 a 3 8 a 4 b 0 4 a 3

Solution

2a 2a

Exercise 70

24 x 4 y 4 z 0 w 8 9xy w 7 24 x 4 y 4 z 0 w 8 9xy w 7

Exercise 71

t 2 ( y 4 ) t 2 ( y 4 )

Solution

t 2 y 4 t 2 y 4

Exercise 72

x 3 ( x 6 x 2 ) x 3 ( x 6 x 2 )

Exercise 73

a 4 b 6 ( a 10 b 16 a 5 b 7 ) a 4 b 6 ( a 10 b 16 a 5 b 7 )

Solution

a 9 b 15 a 9 b 15

Exercise 74

3 a 2 b 3 ( 14 a 2 b 5 2b ) 3 a 2 b 3 ( 14 a 2 b 5 2b )

Exercise 75

(x+3y) 11 (2x−1) 4 (x+3y) 3 (2x−1) (x+3y) 11 (2x−1) 4 (x+3y) 3 (2x−1)

Solution

( x+3y ) 8 ( 2x−1 ) 3 ( x+3y ) 8 ( 2x−1 ) 3

Exercise 76

40 x 5 z 10 (z− x 4 ) 12 (x+z) 2 10 z 7 (z− x 4 ) 5 40 x 5 z 10 (z− x 4 ) 12 (x+z) 2 10 z 7 (z− x 4 ) 5

Exercise 77

x n x r x n x r

Solution

x n+r x n+r

Exercise 78

a x b y c 5z a x b y c 5z

Exercise 79

x n ⋅ x n+3 x n ⋅ x n+3

Solution

x 2n+3 x 2n+3

Exercise 80

x n+3 x n x n+3 x n

Exercise 81

x n+2 x 3 x 4 x n x n+2 x 3 x 4 x n

Solution

x x

Exercise 82

a to the power star, a to the power circle.

Exercise 84

y Δ y ∇ y Δ y ∇

Exercise 85

a Δ a ∇ b □ b ⋄ a Δ a ∇ b □ b ⋄

Solution

a Δ+∇ b □+⋄ a Δ+∇ b □+⋄

Exercises for Review

Exercise 86

((Reference)) What natural numbers can replace x x so that the statement −5<x≤3 −5<x≤3 is true?

Exercise 87

((Reference)) Use the distributive property to expand 4x(2a+3b) 4x(2a+3b) .

Solution

8ax+12bx 8ax+12bx

Exercise 88

((Reference)) Express xxxyyyy(a+b)(a+b) xxxyyyy(a+b)(a+b) using exponents.

Exercise 89

((Reference)) Find the value of 4 2 + 3 2 ⋅ 2 3 −10⋅8 4 2 + 3 2 ⋅ 2 3 −10⋅8 .

Solution

8

Exercise 90

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

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