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Properties of Exponents

Author: Sophia
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what's covered
In this lesson, you will learn how to simplify an expression using the properties of exponents. Specifically, this lesson will cover:

Table of Contents

1. Properties of Exponents

Problems with exponents can often be simplified using a few basic exponent properties. Exponents represent repeated multiplication. We will use this fact to discover important properties.

did you know
The word exponent comes from the Latin "expo" meaning "out of" and "ponere" meaning "place." While there is some debate, it seems that the Babylonians living in present-day Iraq were the first to do work with exponents (dating back to the 23rd century BC or earlier!)

1a. Product Property of Exponents

Let's take a look at the following example and how we can rewrite the exponents as a multiplication problem.

EXAMPLE

a cubed a squared Expand exponents to multiplication problem
open parentheses a a a close parentheses open parentheses a a close parentheses Now we have 5 space a apostrophe s being multiplied together
a to the power of 5 Our Solution

A quicker method to arrive at our answer would have been to just add the exponents. This is known as the product property of exponents.

formula to know
Product Property of Exponents
a to the power of m • a to the power of n equals a to the power of m plus n end exponent

The important thing here is that the expressions must have the same base. If exponential expressions with the same base are multiplied together, we can simply add the exponents.

EXAMPLE

3 squared times 3 to the power of 6 times 3 Same base, add the exponents 2 plus 6 plus 1
3 to the power of 9 Our Solution

1b. Quotient Property of Exponents

Rather than multiplying, we will now try to divide with exponents.

EXAMPLE

a to the power of 5 over a squared Expand Exponents
fraction numerator a a a a a over denominator a a end fraction Divide out two of the a apostrophe s
a a a Convert to Exponents
a cubed Our Solution

A quicker method to arrive at the solution would have been to just subtract the exponents. This is known as the quotient property of exponents:

formula to know
Quotient Property of Exponents
a to the power of m over a to the power of n equals a to the power of left parenthesis m minus n right parenthesis end exponent

Just like with the product property, it is important to note that it only holds true when the bases are the same.

EXAMPLE

7 to the power of 13 over 7 to the power of 5 Same base, subtract the exponents 13 minus 5
7 to the power of 8 Our Solution

1c. Power of a Power Property of Exponents

A third property we will look at will have an exponent raised to another exponent. This is investigated in the following example:

EXAMPLE

open parentheses a squared close parentheses cubed This means we have a squared three times
a squared times a squared times a squared Add exponents
a to the power of 6 Our Solution

A quicker method to arrive at the solution would have been to just multiply the exponents. This is known as the power of a power property of exponents.

formula to know
Power of a Power Property of Exponents
left parenthesis a to the power of m right parenthesis to the power of n equals a to the power of m n end exponent

This property is often combined with two other properties: power of a product, and power of a quotient. We will look at these properties next.

1d. Power of a Product Rule

What happens when you have more than one factor being multiplied together and raised to a power?

EXAMPLE

open parentheses a b close parentheses cubed This means we have open parentheses a b close parentheses three times
open parentheses a b close parentheses open parentheses a b close parentheses open parentheses a b close parentheses Three a apostrophe s and three b apostrophe s can be written with exponents
a cubed b cubed Our Solution

A quicker method to arrive at the solution would have been to take the exponent of three and put it on each factor in parentheses. This is known as the power of a product property of exponents.

formula to know
Power of a Product Property of Exponents
left parenthesis a b right parenthesis to the power of m equals a to the power of m b to the power of m

hint
It is important to be careful to only use the power of a product property with multiplication inside parentheses. This property does NOT work if there is addition or subtraction.

open parentheses a plus b close parentheses to the power of m not equal to a to the power of m plus b to the power of m

These are NOT equal. Beware of this error!

1e. Power of a Quotient Property of Exponents

Now, what about when you are dividing terms and that whole set is raised to a power?

EXAMPLE

open parentheses a over b close parentheses cubed This means we have the fraction three times
open parentheses a over b close parentheses open parentheses fraction numerator begin display style a end style over denominator begin display style b end style end fraction close parentheses open parentheses fraction numerator begin display style a end style over denominator begin display style b end style end fraction close parentheses Multiply fractions across the top and bottom, using exponents
a cubed over b cubed Our Solution

A quicker method to arrive at the solution would have been to put the exponent on every factor in both the numerator and denominator. This is known as the power of a quotient property of exponents.

formula to know
Power of a Quotient Property of Exponents
open parentheses a over b close parentheses to the power of m equals a to the power of m over b to the power of m

2. Putting It All Together

The power of a power, product, and quotient properties of exponents are often used together to simplify expressions.

EXAMPLE

open parentheses a to the power of 5 close parentheses cubed a squared Use the Power of a Power Property to multiply 5 and 3
a to the power of 5 times 3 end exponent a squared Evaluate the multiplication in the exponent
a to the power of 15 a squared Use the Product Property to add the exponents 15 and 2
a to the power of 15 plus 2 end exponent Evaluate the addition in the exponent
a to the power of 17 Our Solution

EXAMPLE

open parentheses b to the power of 7 close parentheses squared over b to the power of 6 Use the Power of a Power Property to multiply 7 and 2
b to the power of 7 times 2 end exponent over b to the power of 6 Evaluate the multiplication in the exponent
b to the power of 14 over b to the power of 6 Use the Quotiet Property to subtract the exponents 14 and 6
b to the power of 14 minus 6 end exponent Evaluate the subtraction in the exponent
b to the power of 8 Our Solution

big idea
Rules of Exponents Formula
Product Rule of Exponents a to the power of m a to the power of n equals a to the power of m plus n end exponent
Quotient Rule of Exponents a to the power of m over a to the power of n equals a to the power of m minus n end exponent
Power of a Power Rule of Exponents open parentheses a to the power of m close parentheses to the power of n equals a to the power of m n end exponent
Power of a Product Rule of Exponents open parentheses a b close parentheses to the power of m equals a to the power of m b to the power of m
Power of a Quotient Rule of Exponents open parentheses a over b close parentheses to the power of m equals a to the power of m over b to the power of m

summary
These five properties of exponents are often mixed up in the same problem. Often there is a bit of flexibility as to which property is used first. However, the order of operations still applies to a problem. For this reason, we suggest simplifying inside any parentheses first, then simplify any exponents (using power properties). Finally, simplify any multiplication or division (using product and quotient properties).

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Formulas to Know
Power of a Power Property of Exponents

open parentheses a to the power of n close parentheses to the power of m equals a to the power of n m end exponent

Power of a Product Property of Exponents

open parentheses a b close parentheses to the power of n equals a to the power of n b to the power of n

Power of a Quotient Property of Exponents

open parentheses a over b close parentheses to the power of n equals a to the power of n over b to the power of n

Product Property of Exponents

a to the power of n times a to the power of m equals a to the power of n plus m end exponent

Quotient Property of Exponents

a to the power of n over a to the power of m equals a to the power of n minus m end exponent

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