5.2: Use Multiplication Properties of Exponents
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Learning Objectives
By the end of this section, you will be able to:
- Simplify expressions with exponents
- Simplify expressions using the Product Property for Exponents
- Simplify expressions using the Power Property for Exponents
- Simplify expressions using the Product to a Power Property
- Simplify expressions by applying several properties
- Multiply monomials
Note
Before you get started, take this readiness quiz.
- Simplify: 34⋅34
If you missed this problem, review Exercise 1.6.13. - Simplify: (−2)(−2)(−2).
If you missed this problem, review Exercise 1.5.13.
Simplify Expressions with Exponents
Remember that an exponent indicates repeated multiplication of the same quantity. For example, 24 means the product of 4 factors of 2, so 24 means 2·2·2·2.
Let’s review the vocabulary for expressions with exponents.
EXPONENTIAL NOTATION
This is read a to the mth power.
In the expression am, the exponent m tells us how many times we use the base a as a factor.
Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.
Example 5.2.1
Simplify:
- 43
- 71
- (56)2
- (0.63)2
- Answer
-
- 43 Multiply three factors of 4.4⋅4⋅4 Simplify. 64
- 71Multiply one factor of 7.7
- (56)2 Multiply two factors. (56)(56) Simplify. 2536
- (0.63)2 Multiply two factors. (0.63)(0.63) Simplify. 0.3969
Example 5.2.2
Simplify:
- 63
- 151
- (37)2
- (0.43)2
- Answer
-
- 216
- 15
- 949
- 0.1849
Example 5.2.3
Simplify:
- 25
- 211
- (25)3
- (0.218)2
- Answer
-
- 32
- 21
- 8125
- 0.047524
Example 5.2.4
Simplify:
- (−5)4
- −54
- Answer
-
- (−5)4 Multiply four factors of −5(−5)(−5)(−5) Simplify. 625
- −54 Multiply four factors of 5.−(5⋅5⋅5⋅5) Simplify. −625
Notice the similarities and differences in Example 5.2.4 part 1 and Example 5.2.4 part 2! Why are the answers different? As we follow the order of operations in part 1 the parentheses tell us to raise the (−5) to the 4th power. In part 2 we raise just the 5 to the 4th power and then take the opposite.
Example 5.2.5
Simplify:
- (−3)4
- −34
- Answer
-
- 81
- −81
Example 5.2.6
Simplify:
- (−13)4
- −134
- Answer
-
- 169
- −169
Simplify Expressions Using the Product Property for Exponents
You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.
We’ll derive the properties of exponents by looking for patterns in several examples.
First, we will look at an example that leads to the Product Property.
![]() |
|
What does this mean? How many factors altogether? |
![]() |
So, we have | ![]() |
Notice that 5 is the sum of the exponents, 2 and 3. | ![]() |
We write: x2⋅x3x2+3x5
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.
PRODUCT PROPERTY FOR EXPONENTS
If a is a real number, and m and n are counting numbers, then
am⋅an=am+n
To multiply with like bases, add the exponents.
An example with numbers helps to verify this property.
23⋅22?=22+34⋅8?=2532=32✓
Example 5.2.7
Simplify: y5⋅y6
- Answer
-
Use the product property, am⋅an=am+n. Simplify.
Example 5.2.8
Simplify: b9⋅b8
- Answer
-
b17
Example 5.2.9
Simplify: x12⋅x4
- Answer
-
x16
Example 5.2.10
Simplify:
- 25⋅29
- 3⋅34
- Answer
-
a.
Use the product property, am⋅an=am+n. Simplify. b.
Use the product property, am⋅an=am+n. Simplify.
Example 5.2.11
Simplify:
- 5⋅55
- 49⋅49
- Answer
-
- 56
- 418
Example 5.2.12
Simplify:
- 76⋅78
- 10⋅1010
- Answer
-
- 714
- 1011
Example 5.2.13
Simplify:
- a7⋅a
- x27⋅x13
- Answer
-
a.
Rewrite, a=a1 Use the product property, am⋅an=am+n. Simplify. b.
Notice, the bases are the same, so add the exponents. Simplify.
Example 5.2.14
Simplify:
- p5⋅p
- y14⋅y29
- Answer
-
- p6
- y43
Example 5.2.15
Simplify:
- z⋅z7
- b15⋅b34
- Answer
-
- z8
- b49
We can extend the Product Property for Exponents to more than two factors.
Example 5.2.16
Simplify: d4⋅d5⋅d2
- Answer
-
Add the exponents, since bases are the same. Simplify.
Example 5.2.17
Simplify: x6⋅x4⋅x8
- Answer
-
x18
Example 5.2.18
Simplify: b5⋅b9⋅b5
- Answer
-
b19
Simplify Expressions Using the Power Property for Exponents
Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.
![]() |
|
What does this mean? How many factors altogether? |
![]() |
So we have | ![]() |
Notice that 6 is the product of the exponents, 2 and 3. | ![]() |
We write:
(x2)3x2⋅3x6
We multiplied the exponents. This leads to the Power Property for Exponents.
POWER PROPERTY FOR EXPONENTS
If a is a real number, and m and n are whole numbers, then
(am)n=am⋅n
To raise a power to a power, multiply the exponents.
An example with numbers helps to verify this property.
(32)3?=32⋅3(9)3?=36729=729✓
Example 5.2.19
Simplify:
- (y5)9
- (44)7
- Answer
-
a.
Use the power property, (am)n=am⋅n. Simplify. b.
Use the power property. Simplify.
Example 5.2.20
Simplify:
- (b7)5
- (54)3
- Answer
-
- b35
- 512
Example 5.2.21
Simplify:
- (z6)9
- (37)7
- Answer
-
- z54
- 349
Simplify Expressions Using the Product to a Power Property
We will now look at an expression containing a product that is raised to a power. Can you find this pattern?
What does this mean? (2x) 3 We group the like factors together. 2x⋅2x⋅2x How many factors of 2 and of x?2⋅2⋅x3 Notice that each factor was raised to the power and (2x)3 is 23⋅x3
We write:(2x)323⋅x3
The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.
PRODUCT TO A POWER PROPERTY FOR EXPONENTS
If a and b are real numbers and m is a whole number, then
(ab)m=ambm
To raise a product to a power, raise each factor to that power.
An example with numbers helps to verify this property:
(2⋅3)2?=22⋅3262?=4⋅936=36✓
Example 5.2.22
Simplify:
- (−9d)2
- (3mn)3.
- Answer
-
a.
Use Power of a Product Property, (ab)m=ambm. Simplify. Use Power of a Product Property, (ab)m=ambm. Simplify.
Example 5.2.23
Simplify:
- (−12y)2
- (2wx)5
- Answer
-
- 144y2
- 32w5x5
Example 5.2.24
Simplify:
- (5wx)3
- (−3y)3
- Answer
-
- 125w3x3
- −27y3
Simplify Expressions by Applying Several Properties
We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.
PROPERTIES OF EXPONENTS
If a and b are real numbers, and m and n are whole numbers, then
Product Property am⋅an=am+nPower Property (am)n=amnProduct to a Power (ab)m=ambm
All exponent properties hold true for any real numbers m and n. Right now, we only use whole number exponents.
Example 5.2.25
Simplify:
- (y3)6(y5)4
- (−6x4y5)2
- Answer
-
- (y3)6(y5)4 Use the Power Property. y18⋅y20 Add the exponents. y38
- (−6x4y5)2 Use the Product to a Power Property. (−6)2(x4)2(y5)2 Use the Power Property. (−6)2(x8)(y10)2 Simplify. 36x8y10
Example 5.2.26
Simplify:
- (a4)5(a7)4
- (−2c4d2)3
- Answer
-
- a48
- −8c12d6
Example 5.2.27
Simplify:
- (−3x6y7)4
- (q4)5(q3)3
- Answer
-
- 81x24y28
- q29
Example 5.2.28
Simplify:
- (5m)2(3m3)
- (3x2y)4(2xy2)3
- Answer
-
- (5m)2(3m3) Raise 5m to the second power. 52m2⋅3m3 Simplify. 25m2⋅3m3 Use the Commutative Property. 25⋅3⋅m2⋅m3 Multiply the constants and add the exponents. 75m5
- (3x2y)4(2xy2)3Use the Product to a Power Property.(34x8y4)(23x3y6)Simplify.(81x8y4)(8x3y6)Use the Commutative Property.81⋅8⋅x8⋅x3⋅y4⋅y6Multiply the constants and add the exponents.648x11y10
Example 5.2.29
Simplify:
- (5n)2(3n10)
- (c4d2)5(3cd5)4
- Answer
-
- 75n12
- 81c24d30
Example 5.2.30
Simplify:
- (a3b2)6(4ab3)4
- (2x)3(5x7)
- Answer
-
- 256a22b24
- 40x10
Multiply Monomials
Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.
Example 5.2.31
Multiply: (3x2)(−4x3)
- Answer
-
(3x2)(−4x3)Use the Commutative Property to rearrange the terms.3⋅(−4)⋅x2⋅x3Multiply.−12x5
Example 5.2.32
Multiply: (5y7)(−7y4)
- Answer
-
−35y11
Example 5.2.33
Multiply: (−6b4)(−9b5)
- Answer
-
54b9
Example 5.2.34
Multiply: (56x3y)(12xy2)
- Answer
-
(56x3y)(12xy2)Use the Commutative Property to rearrange the terms.56⋅12⋅x3⋅x⋅y⋅y2Multiply.10x4y3
Example 5.2.35
Multiply: (25a4b3)(15ab3)
- Answer
-
6a5b6
Example 5.2.36
Multiply: (23r5s)(12r6s7)
- Answer
-
8r11s8
Note
Access these online resources for additional instruction and practice with using multiplication properties of exponents:
Key Concepts
- Exponential Notation
- Properties of Exponents
- If a and b are real numbers and m and n are whole numbers, then
Product Property am⋅an=am+nPower Property (am)n=amnProduct to a Power (ab)m=ambm