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2.6: Rules of Exponents

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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 is any real number and n is a natural number, then
xn=xxxxn factors of x
An exponent records the number of identical factors in a multiplication.

Base Exponent Power

In xn,

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

The term xn is read as "x to the nth."

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:

x2x4=xx2xxxx4=xxxxxx6=x62+4=6

aa2=a1aa2=aaa3=a31+2=3

Product Rule for Exponents

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

xnxm=xn+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 2.6.1

x3x5=x3+5=x8

Example 2.6.2

a6a14=a6+14=a20

Example 2.6.3

y5y=y5y1=y5+1=y6

Example 2.6.4

(x2y)8(x2y)5=(x2y)8+5=(x2y)13

Example 2.6.5

x3y4(xy)3+4

Since the bases are not the same, the product rule does not apply.

Practice Set A

Find each product.

Practice Problem 2.6.1

x2x5

Answer

x2+5=x7

Practice Problem 2.6.2

x9x4

Answer

x9+4=x13

Practice Problem 2.6.3

y6y4

Answer

y6+4=y10

Practice Problem 2.6.4

c12c8

Answer

c12+8=c20

Practice Problem 2.6.5

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

Answer

(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 2.6.6

2x37x5=27x3+5=14x8

We used the commutative and associative properties of mulitplication. In practice, we use these properties "mentally" (as signiffied by the drawing of the box). We don't actually write the second step.

Example 2.6.7

4y36y2=46y3+2=24y5

Example 2.6.8

9a2b6(8ab42b3)=982a2+1b6+4+3=144a3b13

Example 2.6.9

5(a+6)23(a+6)8=53(a+6)2+8=15(a+6)10

Example 2.6.10

4x312y2=48x3y2

Practice Set B

Perform each multiplication in one step.

Practice Problem 2.6.6

3x52x2

Answer

6x7

Practice Problem 2.6.7

6y33y4

Answer

18y7

Practice Problem 2.6.8

4a3b29a2b

Answer

36a5b3

Practice Problem 2.6.9

x44y22x27y6

Answer

56x6y8

Practice Problem 2.6.10

(xy)34(xy)2

Answer

4(xy)5

Practice Problem 2.6.11

8x4y2xx3y5

Answer

8x8y7

Practice Problem 2.6.12

2aaa3(ab2a3)b6ab2

Answer

12a10b5

Practice Problem 2.6.13

anamar

Answer

an+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.

x5x2=xxxxxxx=(xx)xxx(xx)=xxx=x3. Notice that 52=3

a8a3=aaaaaaaaaaa=(aaa)aaaaa(aaa)=aaaaa=a5. Notice that 83=5

Quotient Rule for Exponents

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

xnxm=xnm,x0.

Sample Set C

Find the following quotients. All exponents are natural numbers.

Example 2.6.11

x5x2=x52=x3

Example 2.6.12

27a3b6c23a2bc=9a32b61c21=9ab5c

Example 2.6.13

15x3x=5x
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

Practice Problem 2.6.14

y9y5

Answer

y4

Practice Problem 2.6.15

a7a

Answer

a6

Practice Problem 2.6.16

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

Answer

(x+6)2

Practice Problem 2.6.17

26x4y6z213x2y2z

Answer

2x2y4z

When we make the subtraction, nm, in the division, xnxm, there are three possibilities for the values of the exponents:

The exponent of the numerator is greater than the exponent of the denominator, that is, n>m. Thus, the exponent, nm, is a natural number.

The exponents are the same, that is, n=m. Thus, the exponent, nm, is zero, a whole number.

The exponent of the denominator is greater than the exponent of the numerator, that is, n<m. Thus, the exponent, nm, 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, the subtraction nn produces 0.

Thus, by the second rule of exponents, xnxn=xnn=x0

But what real number, if any, does x0 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.

88=1, 4343=1, 258258=1

Since the letter x represents some nonzero real number divided by itself. Then xnxn=1.

But we have also established that if x0, xnxn=x0. We now have that xnxn=x0 and xnxn=1. This implies that x0=1, x0.

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 x0, x0=1

Any number, other than 0, raised to the power of 0, is 1. 00 has no meaning (it does not represent a number).

Sample Set D

Find each value. Assume the base is not zero.

Example 2.6.14

60=1

Example 2.6.15

2460=1

Example 2.6.16

(2a+5)0=1

Example 2.6.17

4y0=41=4

Example 2.6.18

y6y6=y0=1

Example 2.6.19

2x2x2=2x0=21=2

Example 2.6.20

5(x+4)8(x1)55(x+4)3(x1)5=(x+4)83(x1)55=(x+4)5(x1)0=(x+4)5

Practice Set D

Find each value. Assume the base is not zero.

Practice Problem 2.6.18

y7y3

Answer

y73=y4

Practice Problem 2.6.19

6x42x3

Answer

3x43=3x

Practice Problem 2.6.20

14a77a2

Answer

2a72=2a5

Practice Problem 2.6.21

26x2y54xy2

Answer

132xy3

Practice Problem 2.6.22

36a4b3c88ab3c6

Answer

92a3c2

Practice Problem 2.6.23

51(a4)317(a4)

Answer

3(a4)2

Practice Problem 2.6.24

52a7b3(a+b)826a2b(a+b)8

Answer

2a5b2

Practice Problem 2.6.25

ana3

Answer

an3

Practice Problem 2.6.26

14xrypzq2xryhz5

Answer

7yphzq5

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 2.6.1

3233

Answer

35=243

Exercise 2.6.2

5254

Exercise 2.6.3

9092

Answer

92=81

Exercise 2.6.4

7370

Exercise 2.6.5

2425

Answer

29=512

Exercise 2.6.6

x5x4

Exercise 2.6.7

x2x3

Answer

x5

Exercise 2.6.8

a9a7

Exercise 2.6.9

y5y7

Answer

y12

Exercise 2.6.10

m10m2

Exercise 2.6.11

k8k3

Answer

k11

Exercise 2.6.12

y3y4y6

Exercise 2.6.13

3x22x5

Answer

6x7

Exercise 2.6.14

a2a3a8

Exercise 2.6.15

4y45y6

Answer

20y10

Exercise 2.6.16

2a3b23ab

Exercise 2.6.17

12xy3z24x2y2z3x

Answer

144x4y5z3

Exercise 2.6.18

(3ab)(2a2b)

Exercise 2.6.19

(4x2)(8xy3)

Answer

32x3y3

Exercise 2.6.20

(2xy)(3y)(4x2y5)

Exercise 2.6.21

(14a2b4)(12b4)

Answer

18a2b8

Exercise 2.6.22

(38)(1621x2y3)(x3y2)

Exercise 2.6.23

8583

Answer

82=64

Exercise 2.6.24

6463

Exercise 2.6.25

2924

Answer

25=32

Exercise 2.6.26

416413

Exercise 2.6.27

x5x3

Answer

x2

Exercise 2.6.28

y4y3

Exercise 2.6.29

y9y4

Answer

y5

Exercise 2.6.30

k16k13

Exercise 2.6.31

x4x2

Answer

x2

Exercise 2.6.32

y5y2

Exercise 2.6.33

m16m9

Answer

m7

Exercise 2.6.34

a9b6a5b2

Exercise 2.6.35

y3w10yw5

Answer

y2w5

Exercise 2.6.36

m17n12m16n10

Exercise 2.6.37

x5y7x3y4

Answer

x2y3

Exercise 2.6.38

15x20y24z45x19yz

Exercise 2.6.39

e11e11

Answer

e0=1

Exercise 2.6.40

6r46r4

Exercise 2.6.41

x0x0

Answer

x0=1

Exercise 2.6.42

a0b0c0

Exercise 2.6.43

8a4b04a3

Answer

2a

Exercise 2.6.44

24x4y4z0w89xyw7

Exercise 2.6.45

t2(y4)

Answer

t2y4

Exercise 2.6.46

x3(x6x3)

Exercise 2.6.47

a4b6(a10b16a5b7)

Answer

a9b15

Exercise 2.6.48

3a2b3(14a2b52b)

Exercise 2.6.49

(x+3y)11(2x1)4(x+3y)3(2x1)

Answer

(x+3y)8(2x1)3

Exercise 2.6.50

40x5z10(zx4)12(x+z)210z7(zx4)5

Exercise 2.6.51

xnxr

Answer

xn+r

Exercise 2.6.52

axbyz5z

Exercise 2.6.53

xnxn+3

Answer

x2n+3

Exercise 2.6.54

xn+3xn

Exercise 2.6.55

xn+2x3x4xn

Answer

x

Exercises for Review

Exercise 2.6.56

What natural numbers can replace x so that the statement 5<x3 is true?

Exercise 2.6.57

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

Answer

8ax+12bx

Exercise 2.6.58

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

Exercise 2.6.59

Find the value of 42+3223108

Answer

8

Exercise 2.6.60

Find the value of 42+(3+2)21235+24(3223+42.


This page titled 2.6: Rules of Exponents is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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