Simplifying Variable Expressions Using Exponent Properties

Learning Outcomes

  • Use the product property of exponents to simplify expressions
  • Use the power property of exponents to simplify expressions
  • Use the product to a power property of exponents to simplify expressions

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. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.

First, we will look at an example that leads to the Product Property.

x2x3
What does this mean?

How many factors altogether?

.
So, we have xxxxx=x5
Notice that 5 is the sum of the exponents, 2 and 3. x2x3 is x2+3, or x5
We write: x2x3

x2+3

x5

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Product Property of Exponents

If a is a real number and m,n are counting numbers, then

aman=am+n
To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

2223=?22+348=?2532=32

example

Simplify: x5x7

Solution

x5x7
Use the product property, aman=am+n. x5+7
Simplify. x12

 

try it

 

example

Simplify: b4b

 

try it

 

example

Simplify: 2729

 

try it

 

example

Simplify: y17y23

 

try it

We can extend the Product Property of Exponents to more than two factors.

example

Simplify: x3x4x2

 

try it

In the following video we show more examples of how to use the product rule for exponents to simplify expressions.

Simplify Expressions Using the Power Property of 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.

(x2)3
x2x2x2
What does this mean?

How many factors altogether?

.
So, we have xxxxxx=x6
Notice that 6 is the product of the exponents, 2 and 3. (x2)3 is x23 or x6
We write: (x2)3

x23

x6

We multiplied the exponents. This leads to the Power Property for Exponents.

Power Property of Exponents

If a is a real number and m,n are whole numbers, then

(am)n=amn
To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

(52)3=?523(25)3=?5615,625=15,625

example

Simplify:

1. (x5)7
2. (36)8

 

try it

Watch the following video to see more examples of how to use the power rule for exponents to simplify expressions.

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. Look for a pattern.

(2x)3
What does this mean? 2x2x2x
We group the like factors together. 222xxx
How many factors of 2 and of x? 23x3
Notice that each factor was raised to the power. (2x)3 is 23x3
We write: (2x)3

23x3

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

Product to a Power Property of 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:

(23)2=?223262=?4936=36

example

Simplify: (11x)2

 

try it

 

example

Simplify: (3xy)3

 

try it

In the next video we show more examples of how to simplify a product raised to a power.