1.1.4: Integer Exponents

Learning Objectives

By the end of this section, you will be able to: 

• Understand the meaning of an integer exponent.
• Simplify numerical expressions with which involve integer exponents.

Be Prepared

Before we get started, take this readiness quiz.

1. Evaluate −5 ⋅2.

2. Evaluate −5 +3.

3. Evaluate 3 ⋅3 ⋅3 ⋅3.

We will begin with positive integer exponents where the meaning is straight forward.  The positive integer exponent indicates repeated multiplication of the same quantity. For example, in the expression 𝑎𝑚, the positive integer exponent 𝑚 tells us how many times we use the base 𝑎 as a factor.

𝑎𝑚 =𝑎⋅𝑎⋅⋯⋅𝑎⏟__⏟__⏟𝑚⁢𝑎′⁢s

For example,

(−9)5 =(−9)⁢(−9)⁢(−9)⁢(−9)⁢(−9).

Let’s review the vocabulary for expressions with exponents.

Definition 1.1⁢.4.1

                                                                                     

                                                                            𝑎𝑚 =𝑎⋅𝑎⋯𝑎⏟𝑚⁢𝑎′⁢s

This is read 𝑎 to the 𝑚th (power), or 𝑎 to the power 𝑚.

In the expression 𝑎𝑚 with positive integer 𝑚 and 𝑎 ≠0, the exponent 𝑚 tells us how many times we use the base 𝑎 as a factor.

Example 1.1⁢.4.2

a. Evaluate 23.

b. Evaluate −72.

c. Evaluate (−1)4.

d. Write using exponents: −2 ⋅2 ⋅2.

e. Identify the base and the exponent: −43.

Solution

a. 23 =2 ⋅2 ⋅2 =8

b. −72 =−49

c. (−1)4 =(−1) ⋅(−1) ⋅(−1) ⋅(−1) =1

d. −2 ⋅2 ⋅2 =−23

e. −43 has an exponent of 3 and the base is 4 since there are no parentheses that indicate including the "-".

Try It 1.1⁢.4.3

a. Evaluate 34.

b. Evaluate −24.

c. Evaluate (−2)3.

d. Write using exponents −6 ⋅6 ⋅6 ⋅6.

e. Identify the base and the exponent: −2 ⋅57.

Answer

a. 81

b. -16

c. -8

d. −64

e. The exponent is 7 and the base is 5 (since there are no parentheses that would include 2 or -2).

We will now investigate several properties of exponents.   First we will look at an example that leads to the Product Property for Positive Integer Exponents.

	72⁢73
What does this mean?	=7⋅7⏟2 factors  ⋅7⋅7⋅7⏟3 factors =7⋅7⋅7⋅7⋅7⏟__⏟__⏟5 factors
	=75

The base stayed the same and we added the exponents.  Remember that the exponent counts the number of bases we multiply so, we are multiplying 2 sevens by 3 sevens which gives us a total of 5 sevens!

 

In general we have:

Product property for positive integer exponents

If 𝑎 is a real number and 𝑚 and 𝑛 are positive integers, then

𝑎𝑚⁢𝑎𝑛=𝑎𝑚+𝑛.

To multiply with like bases, add the exponents.

Example 1.1⁢.4.4

Simplify each expression:

a. 35⁢36

b. 24 ⋅23⋅4

c. 2 ⋅57 ⋅3 ⋅5

d. 114⁢115⁢112

Solution

a.

	35⁢36
Use the Product Property, 𝑎𝑚⁢𝑎𝑛 =𝑎𝑚+𝑛 or directly use the fact that the exponent counts.	=35+6
Simplify.	=311

b.

	24 ⋅23⋅4
Use the Product Property, 𝑎𝑚⁢𝑎𝑛 =𝑎𝑚+𝑛 or or directly use the fact that the exponent counts..	=24+3⋅4
Simplify.	=24+12 =216

c.

	2 ⋅57 ⋅3 ⋅5
Rewrite, 𝑎 =𝑎1.	=2 ⋅57 ⋅3 ⋅51
Use the Commutative Property and use the Product Property, 𝑎𝑚⁢𝑎𝑛 =𝑎𝑚+𝑛 or or directly use the fact that the exponent counts..	=2 ⋅3 ⋅57+1
Simplify.	=6 ⋅58

d.

	114⁢115⁢112
Add the exponents, since the bases are the same.	=114+5+2
Simplify.	=1111

Try It 1.1⁢.4.5

Simplify each expression:

a. 79⁢78

b. 42⋅3 ⋅43

c. 3 ⋅95 ⋅4 ⋅9

d. 36⁢34⁢38

Answer

a. 717

b. 49

c. 12 ⋅96

d. 318

Try It 1.1⁢.4.6

Simplify each expression:

a. (−2)12⁢(−2)4

b. 10 ⋅105

c. 2 ⋅4 ⋅6 ⋅47

d. 65⁢69⁢65

Answer

a. (−2)16 =216

b. 106

c. 3 ⋅49

d. 619

 

Now let’s look at an exponential expression that contains a power raised to a power. Let's see if we can discover a general property.

	(52)3
What does this mean?	=52⁢52⁢52
How many factors altogether?	=5⋅5⏟2 factors  ⋅5⋅5⏟2 factors  ⋅5⋅5⏟2 factors  =5⋅5⋅5⋅5⋅5⋅5⏟___⏟___⏟6 factors
So we have	=56

Notice the 6 is the product of the exponents, 2 and 3. We see that (52)3 is 52⋅3 or 𝑥6. We can also see that

In this example we multiplied the exponents.

We can check various examples to see that this leads us to the Power Property for Positive Integer Exponents.

Power Property for Integer Exponents

If 𝑎 is a real number and 𝑚 and 𝑛 are positive integers, then

(𝑎𝑚)𝑛=𝑎𝑚⁢𝑛.

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

Example 1.1⁢.4.7

Simplify each expression:

a. (35)9

b. (44)7

c. (23)6⁢(25)4

Solution

a.

	(35)9
Use the power property, (𝑎𝑚)𝑛 =𝑎𝑚⁢𝑛 or use the fact that the exponent counts the number of bases being multiplied.	35⋅9
Simplify.	345

b.

	(44)7
Use the power property.	=44⋅7
Simplify.	=428

c.

	(23)6⁢(25)4
Use the power property.	=218⁢220
Add the exponents.	=238

 

Try It 1.1⁢.4.8

Simplify each expression:

a. ((−3)7)5

b. (54)3

c. (34)5⁢(37)4

Answer

a. (−3)35 =−335

b. 512

c. 348

Try It 1.1⁢.4.9

Simplify each expression:

a. (96)9

b. (37)7

c. ((12)4)5⁢((12)3)3

Answer

a. 954

b. 349

c. (12)29

 

We will now look at an expression containing a product that is raised to a power. Can we find this pattern?

	(2⋅5)3
What does this mean?	=2 ⋅5 ⋅2 ⋅5 ⋅2 ⋅5
We group the like factors together.	=2 ⋅2 ⋅2 ⋅5 ⋅5 ⋅5
How many factors of 2 and of 5?	=23⁢53

Notice that each factor was raised to the power and (2⋅5)3 is 23⁢53.

The exponent applies to each of the factors! We can say that the exponent distributes over multiplication. If we were to check various examples with exponents we would find the same pattern emerges. This leads to the Product to a Power Property for Postive Integer Exponents.

Product to a Power Property for Integer Exponents

If 𝑎 and 𝑏 are real numbers and 𝑚 is a positive integer, then

(𝑎⁢𝑏)𝑚=𝑎𝑚⁢𝑏𝑚.

To raise a product to a power, raise each factor to that power.

Example 1.1⁢.4.10

Simplify each expression using the Product to a Power Property:

a. (−3⋅2)3

b. (6⋅43)2

c. (5⋅43)2

Solution

a.

	(−3⋅2)3
Use Power of a Product Property, (𝑎⁢𝑏)𝑚 =𝑎𝑚⁢𝑏𝑚.	=(−3)3⁢23
Simplify.	=−27 ⋅8 =−216

b.

	(6⋅43)2
Use the Power of a Product Property, (𝑎⁢𝑏)𝑚 =𝑎𝑚⁢𝑏𝑚.	=62⁢(43)2
Use the Power Property, (𝑎𝑚)𝑛 =𝑎𝑚⁢𝑛.	=62⁢46
Simplify.	=36 ⋅46

c.

	(5⋅43)2
Use the power of a product property, (𝑎⁢𝑏)𝑚 =𝑎𝑚⁢𝑏𝑚.	=52⁢(43)2
Simplify.	=25 ⋅46

Try It 1.1⁢.4.11

Simplify each expression using the Product to a Power Property:

a. (2⋅3)5

b. (2⋅(−2)3)4

c. (8⁢(−2)4)2

Answer

a. 32 ⋅35

b. 16⁢(−2)12 =16 ⋅212 =216

c. 64⁢(−2)8 =64 ⋅28 =214

Try It 1.1⁢.4.12

Simplify each expression using the Product to a Power Property:

a. (−3⋅2)3

b. (−4⋅104)2

c. (2⋅104)3

Answer

a. −27 ⋅23 =−27 ⋅8 =−216

b. 16 ⋅108

c. 8 ⋅1012

Because division is multiplication by a recipricol, the exponent must distribute over division as well as multiplication.  We give a specific example here:

(43)3=43⋅43⋅43=4333.

So we have also the quotient property.

Quotient to a Power Property for Integer Exponents

If 𝑎 and 𝑏( ≠0 are real numbers and 𝑚 is a positive integer, then

(𝑎𝑏)𝑚=𝑎𝑚𝑏𝑚.

To raise a product to a power, raise each factor to that power.

Example 1.1⁢.4.13

Rewrite (35)3 using the Quotient to a Power property

Solution

(35)3 =3353.

Try It 1.1⁢.4.14

Rewrite (12)5 using the Quotient to a Power property

Answer

125.

Try It 1.1⁢.4.15

Rewrite (−23)4 using the Quotient to a Power property

Answer

(−2)434.  This is equal to 1681 but this isn't what is asked here.

Now we will extend the meaning to all integers.  

Extending the Meaning of Exponent to Integers

Note that, while so far an exponent that is not a positive integer has no meaning, we see that blindly applying the above properties for such exponents leads to a couple definitions.

To give an idea of the argument we begin with a specific example.   So far we do not have a meaning for 70 nor for 7−3 since the exponents in these examples are not counting numbers. We will assume the rules of exponents apply because we would like the meaning of integer exponents to be consistent with these rules. 

We know 

	70⁢71
Using the addition of exponents.	=70+1
Simplify the exponent.	=71
Conclude.	70⁢7 =7

70 is the number that when you multiply it by 7 the result is 7.	70 =1.

We could replace 7 with any non-zero number and have the same conclusion!  So any non-zero number to the zeroth power is 1.

So, in general,  if the Product Property is also to hold for the exponent zero we must define, for 𝑎 ≠0,

𝑎0=1.

Now, so far, 7−3 has no meaning.  But if it did, and that meaning is consistent with the properties of exponents, then, for example

	7−3⁢73
Blindly applying the product property.	=7−3+3
Simplify exponent.	=70
Using our new definition: 𝑥0 =1.	=1
Draw conclusion.	7−3⁢73 =1
Note the property of the reciprocal.	7−3 is the reciprocal of 73.
Rewrite in symbols.	7−3 =173

In the argument above, we can replace 7 with any non-zero number and 3 with any positive counting number and −3 with its opposite and arrive at a similar conclustion.  So a negative exponent indicates a recipricol of the expression where the exponent is positive.

So, we must define

Definition 1.1⁢.4.16

For 𝑎 any non-zero real number

𝑎0=1

and for 𝑚 any positive integer

𝑎−𝑚=1𝑎𝑚 or, equivalently, 1𝑎−𝑚=𝑎𝑚.

 

Example 1.1⁢.4.17

Simplify each expression:

a. 90

b. (−2)0

c. (−4⋅52)0

d. −30

Solution

The definition says any non-zero number raised to the zero power is 1.

a. Use the definition of the zero exponent. 90 =1

b. Use the definition of the zero exponent. (−2)0 =1

c. Anything raised to the power zero is 1. Here the base is −4 ⋅52, so (−4⋅52)0 =1

d. Anything raised to the power zero is 1. Here the base is 3, so this is the opposite of 30, or, the opposite of 1. So, −30 =−1

 

Try It 1.1⁢.4.18

Simplify each expression:

a. 110

b. (−620)0

c. (−12⋅53⁢82)0

d. −70

Answer

a. 1

b. 1

c. 1

d. -1

Try It 1.1⁢.4.19

Simplify each expression:

a. 230

b. 𝜋0

c. (2⁢(−27)5)0

d. −170

Answer

a. 1

b. 1

c. 1

d. -1

Example 1.1⁢.4.20

Simplify each expression. Write your answer using positive exponents.

a. 2−5

b. 10−3

c. 13−4

d. 13−2

Solution

a.

	2−5
Use the definition of a negative exponent, 𝑎−𝑛 =1𝑎𝑛.	=125 =132

b.

	10−3
Use the definition of a negative exponent, 𝑎−𝑛 =1𝑎𝑛.	=1103
Simplify.	=11000

c.

	13−4
Use the definition of a negative exponent, 1𝑎−𝑛 =𝑎𝑛.	=34 =81

d.

	13−2
Use the definition of a negative exponent, 1𝑎−𝑛 =𝑎𝑛.	=32
Simplify.	=9

 

Try It 1.1⁢.4.21

Simplify each expression. Write your answer using positive exponents.

a. 2−3

b. 10−7

c. 17−8

d. 14−3

Answer

a. 123 =18

b. 110,000,000

c. 78 

d. 64

Try It 1.1⁢.4.22

Simplify each expression. Write your answer using positive exponents.

a. 3−2

b. 10−4

c. 1(−2)⁢2−7

d. 12−4

Answer

a. 19

b. 110,000

c. −26 =−64

d. 16

Properties of Negative Exponents

The negative exponent tells us we can rewrite the expression by taking the reciprocal of the base and then changing the sign of the exponent.

Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.

For example, if after simplifying an expression we end up with the expression 𝑥−3, we will take one more step and write 1𝑥3. The answer is considered to be in simplest form when it has only positive exponents.

Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

	(34)−2
Use the definition of a negative exponent, 𝑎−𝑛 =1𝑎𝑛.	=1(34)2
Simplify the denominator.	=1916
Simplify the complex fraction.	=169
But we know that	169 =(43)2
This tells us that	(34)−2 =(43)2

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Integer Exponent Property.

Quotient to a Negative Integer Exponent Property

If 𝑎 and 𝑏 are real numbers, 𝑎 ≠0, 𝑏 ≠0, and 𝑛 is an integer, then

(𝑎𝑏)−𝑛=(𝑏𝑎)𝑛.

Example 1.1⁢.4.23

Simplify each expression. Write your answer using positive exponents.

 (57)−2

 

Solution

 

	(57)−2
Use the Quotient to a Negative Integer Exponent Property, (𝑎𝑏)−𝑛 =(𝑏𝑎)𝑛.
Take the reciprocal of the fraction and change the sign of the exponent.	=(75)2
Simplify.	=4925

 

Try It 1.1⁢.4.24

Simplify each expression. Write your answer using positive exponents.

(23)−4

Answer

8116

Try It 1.1⁢.4.25

Simplify each expression. Write your answer using positive exponents.

 (35)−3

Answer

12527

We would like to verify that the properties of positive integer exponents can be extended to all integer exponents we will postpone this until a later section when we review these properties again in the chapter on polynomials where we will use of variables freely.  Though we will postpone the demonstration, it turns out that all properties of exponents that are valid for all counting numbers are also valid for integers.  And they are also valid for rational numbers, if such exponents are defined appropriately (we have no meaning at this time for rational exponents in general).  We will treat this under the topic of radical expressions.

 

We provide for reference a table of properties of exponents.

Definition	Description
Definition of Zero Exponent	𝑎0 =1,𝑎 ≠0
Definition of Negative Exponents	𝑎−𝑛 =1𝑎𝑛, or equivalently, 1𝑎−𝑛 =𝑎𝑛

Property	Description
Product Property	𝑎𝑚 ·𝑎𝑛 =𝑎𝑚+𝑛
Power Property	(𝑎𝑚)𝑛 =𝑎𝑚·𝑛
Product to a Power	(𝑎⁢𝑏)𝑛 =𝑎𝑛⁢𝑏𝑛
Quotient Property	𝑎𝑚𝑎𝑛 =𝑎𝑚−𝑛,𝑎 ≠0
Quotient to a Power Property	(𝑎𝑏)𝑚 =𝑎𝑚𝑏𝑚,𝑏 ≠0
Quotient to a Negative Exponent	(𝑎𝑏)−𝑛 =(𝑏𝑎)𝑛

 

Example 1.1⁢.4.26

Simplify each expression by applying appropriate properties:

a. (2.5 ×108) ⋅(2 ×104)

b. (25)−2 +70 −(23)2

c. (2⋅5⁢(−3)253⁢(−3)−2)2⁢(12⋅5⁢(−3)353⁢(−3)−1)−1

Solution

a.

	(2.5 ×108) ⋅(2 ×104)
Rearrange product (2.5 ⋅2)⁢𝑐⁢𝑑⁢𝑜⁢𝑡⁢108 ⋅104	(2.5 ⋅2) ⋅108 ⋅104
Simplify.	=5 ⋅108⁢104
Use product property of exponents.	=5 ⋅1012
Rewrite in original format.	=5 ×1012

b.

	(25)−2 +70 −(23)2
Use the quotient to a negative exponent property and the definition of exponent 0.	(52)2 +1 −(23)2
Use the meaning of the positive integer exponent.	=52⁢52 +1 −23⁢23
Multiply fractions.	=254 +1 −49
Add/Subtract fractions (find common denominator) and reduce if possible.	=225+36−1636 =24536

c.

	(2⋅5⁢(−3)253⁢(−3)−2)2⁢(12⋅5⁢(−3)353⁢(−3)−1)−1
Simplify inside the parentheses first: use the meaning of a negative exponent and the product property.	=(2⁢(−3)2⁢(−3)252)2⁢(12⁢(−3)3⁢(−3)152)−1 =(2⁢(−3)452)2⁢(12⁢(−3)452)−1
Use the quotient to a negative exponent property	=(2⁢(−3)452)2⁢(5212⁢(−3)4)1
Use meaning of the exponent or the quotient property and  the Product to a Power Property, (𝑎⁢𝑏)𝑚 =𝑎𝑚⁢𝑏𝑚.	=((2⁢(−3)4)2(52)2) ⋅(5212⁢(−3)4) =(22⁢((−3)4)2(52)2) ⋅(5212⁢(−3)4)
Use the power property.	=(4⁢(−3)8(54) ⋅(5212⁢(−3)4)
Multiply fractions.	=(4⁢(−3)8⁢5212⁢(−3)4⁢54)
Simplify.	=(−3)43⋅52 =3352 =2725

 

Try It 1.1⁢.4.27

Simplify each expression by applying appropriate properties:

a. (9 ×107) ÷(3 ×10−4)

b. (32)−3 −20 −(12)−2

c. (3⁢(−2)⁢52(−2)2⁢(5)−3)2

Answer

a. 3 ×1011

b. −22727

c. 951022 =9⋅5104 To find the actual value you might want to use a calculator.

Try It 1.1⁢.4.28

Simplify each expression by applying appropriate properties:

a. 8×10−32×10−7

b. (23)−1 +2−2 −(23)0

c. (4⋅23⁢3222⁢3−1)2⁢(8⋅2⋅3−322⋅3)−1

Answer

a. 4 ×104

b. 34

c. 24⁢37.  To find the value you might want to use a calculator.

Writing Exercises 1.1⁢.4.29

1. How is the negative exponent related to reciprocals? Give an example.
2. How are positive and negative exponents used in science to express large or small numbers?
3. What is the purpose in writing numbers this way?

Exit Problem 

a. Simplify (3⋅5−3(−2)−5)−3. Write your final answer with positive exponents only.

b. Simplify 36⁢(−3)5⁢21070⁢(−3)15⁢25. Write your final answer with positive exponents only.

Key Concepts

• Exponential Notation
  
  This is read 𝑎 to the 𝑚th power.
  In the expression 𝑎𝑚, the exponent 𝑚 (when positive) tells us how many times we use the base 𝑎 as a factor.
• Zero Exponent (Definition) If 𝑎 is a non-zero number, then 𝑎0 =1.
• Negative Exponent (Definition) If 𝑛 is an integer and 𝑎 ≠0, then 𝑎−𝑛 =1𝑎𝑛 or, equivalently, 1𝑎−𝑛 =𝑎𝑛.
• Product Property for Exponents
  If 𝑎 is a real number and 𝑚 and 𝑛 are integers, then

  𝑎𝑚⁢𝑎𝑛=𝑎𝑚+𝑛

  To multiply with like bases, add the exponents.
• Quotient Property for Exponents
  If 𝑎 is a real number, 𝑎 ≠0, and 𝑚 and 𝑛 are integers, then

  𝑎𝑚𝑎𝑛=𝑎𝑚−𝑛,𝑚>𝑛and𝑎𝑚𝑎𝑛=1𝑎𝑛−𝑚,𝑛>𝑚

• Quotient to a Negative Exponent Property
  If 𝑎 and 𝑏 are real numbers, 𝑎 ≠0, 𝑏 ≠0 and 𝑛 is an integer, then

  (𝑎𝑏)−𝑛=(𝑏𝑎)𝑛

• Power Property for Exponents
  If 𝑎 is a real number and 𝑚 and 𝑛 are integers, then

  (𝑎𝑚)𝑛=𝑎𝑚⁢𝑛

  To raise a power to a power, multiply the exponents.
• Product to a Power Property for Exponents
  If 𝑎 and 𝑏 are real numbers and 𝑚 is a whole number, then

  (𝑎⁢𝑏)𝑚=𝑎𝑚⁢𝑏𝑚

  To raise a product to a power, raise each factor to that power.
• Quotient to a Power Property for Exponents
  If 𝑎 and 𝑏 are real numbers, 𝑏 ≠0, and 𝑚 is an integer, then

  (𝑎𝑏)𝑚=𝑎𝑚𝑏𝑚

  To raise a fraction to a power, raise the numerator and denominator to that power.
• Summary of Exponent Properties
  If 𝑎 and 𝑏 are real numbers, and 𝑚 and 𝑛 are integers, then

• Property	Description
  Definition of Zero Exponent	𝑎0 =1,𝑎 ≠0
  Definition of Negative Exponents	𝑎−𝑛 =1𝑎𝑛, or equivalently, 1𝑎−𝑛 =𝑎𝑛

• Property	Description
  Product Property	𝑎𝑚 ·𝑎𝑛 =𝑎𝑚+𝑛
  Power Property	(𝑎𝑚)𝑛 =𝑎𝑚·𝑛
  Product to a Power	(𝑎⁢𝑏)𝑛 =𝑎𝑛⁢𝑏𝑛
  Quotient Property	𝑎𝑚𝑎𝑛 =𝑎𝑚−𝑛,𝑎 ≠0
  Quotient to a Power Property	(𝑎𝑏)𝑚 =𝑎𝑚𝑏𝑚,𝑏 ≠0
  Quotient to a Negative Exponent	(𝑎𝑏)−𝑛 =(𝑏𝑎)𝑛