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Properties of Exponents

Properties of Exponents

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Simplify an expression using the properties of exponents.

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Tutorial
what's covered
  1. Properties of Exponents
    1. Product Property
    2. Product Property
    3. Quotient Property
    4. Power of Power Property
    5. Power of a Quotient Property

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


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
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 add the exponents. Here is another 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.

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
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 is only holds true when the bases are the same. Here is an example:

7 to the power of 13 over 7 to the power of 5
Same base, subtract the exponents
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:

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
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

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
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. (a+b)m ≠ am + bm These are NOT equal. Beware of this error!

1e. Power of a Quotient Property of Exponents

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
Power of a Quotient Property of Exponents
left parenthesis a over b right parenthesis to the power of m equals a to the power of m over b to the power of m

The power of a power, product, and quotient properties of exponents are often used together to simplify expressions. This is shown in the following examples:

open parentheses x cubed y z squared close parentheses to the power of 4
Put the exponent of 4 on each factor, multiplying powers
x to the power of 12 y to the power of 4 z to the power of 8
Our Solution


open parentheses fraction numerator a cubed b over denominator c to the power of 8 d to the power of 5 end fraction close parentheses squared
Put the exponent of 2 on each factor, multiplying powers
fraction numerator a to the power of 6 b squared over denominator c to the power of 16 d to the power of 10 end fraction
Our Solution


Rules of Exponents
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: http://wallace.ccfaculty.org/book/book.html

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