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College Algebra Text Tutorial

College Algebra Text Tutorial

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Author: Anthony Varela
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Test tutorial for text version.

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Tutorial

What's Covered

WHAT'S COVERED

Overview

Before You Start

Properties of Exponents

Summary

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Answer Key

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 the important properties.

[DID YOU KNOW ICON]

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 Iraq were the first to do work with exponents (dating back to the 23rd century BC or earlier!)

 

Product of Powers Property

Let's consider multiplication with exponents when the bases are the same:

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

 

KEY FORMULA P r o d u c t space P r o p e r t y space o f space E x p o n e n t s
a to the power of m • a to the power of n equals a to the power of m plus n end exponent

 

The product of powers property can be used to simplify many problems. If exponential expressions with the same base are multiplied, we can add the exponents.  Here is an example:

 

Quotient of Powers Property Exponents

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

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

 

KEY FORMULAQ u o t i e n t space o f space P o w e r s space P r o p e r t y
a to the power of n over a to the power of m equals a to the power of left parenthesis n minus m right parenthesis end exponent

 

The quotient of powers property can similarly be used to simplify exponent problems by subtracting exponents on like-variables. Here is an example:



Power of Power Property

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


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.

 

KEY FORMULAP o w e r space o f space a space P o w e r space P r o p e r t y
left parenthesis a to the power of n right parenthesis to the power of m space equals space a to the power of left parenthesis n m right parenthesis end exponent

This property is often combined with two other properties: power of a product, and power of a quotient.


Power of a Product Property

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

 

KEY FORMULAP o w e r space o f space a space P r o d u c t space P r o p e r t y
left parenthesis a b right parenthesis to the power of n equals a to the power of n • b to the power of n

 

HINT

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


left parenthesis a plus b right parenthesis to the power of n space not equal to space a to the power of n space plus space b to the power of n  

These are NOT equal.  Beware of this error!

 

Power of a Quotient Property

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.

 

KEY FORMULA
P o w e r space o f space a space Q u o t i e n t space P r o p e r t y
left parenthesis a over b right parenthesis to the power of n space equals space space a to the power of n over b to the power of n

 

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



 

 

BIG IDEA


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