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3 Tutorials that teach Properties of Exponents

# Properties of Exponents

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Author: Sophia Tutorial
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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

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:

 Same base, add the exponents Our Solution

1b. Quotient Property of Exponents

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

 Expand Exponents Divide out two of the Convert to Exponents 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

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:

 Same base, subtract the exponents 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:

 This means we have three times Add exponents 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

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

 This means we have three times Three and three can be written with exponents 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
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

 This means we have the fraction three times Multiply fractions across the top and bottom, using exponents 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

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:

 Put the exponent of on each factor, multiplying powers Our Solution Put the exponent of on each factor, multiplying powers Our Solution

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

Power of a Product Property of Exponents

Power of a Quotient Property of Exponents

Product Property of Exponents

Quotient Property of Exponents

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