Learning Objectives

1. Use the definition of exponents.
2. Simplify exponential expressions involving multiplying like bases, zeroas an exponent, dividing like bases, negative exponents, raising a baseto two exponents, raising a product to an exponent and raising a quotientto an exponent. 

Introduction

Tutorial

The exponent tells you how many times a base appears in a PRODUCT.
 
 

Example1: Evaluate .
 

It may look alike, but they ARE NOT exactly the same.  Can yousee the difference between the two??  Hopefully, you noticed thatin example 1, there was a ( ) around the - and the 3.  In this problem,there is no -.  This means the - is NOT part of the base, so it willnot get expanded like it did in example 1. 

It is interpreted as finding the negative or opposite of 3 to thefourth power.

Note that 2 + 3 = 5, which is the exponent we ended up with.  Wehad 2 xs written in a product plus another3 xs written in the product for a total of5 xs in the product.  To indicate thatwe put the 5 in the exponent.
 

Let's put this idea together into a general rule:

The reason is, exponents count how many of your base you have ina product.  So if you are continuing that product, you are addingon to the exponents.
 

Example4:   Use the product rule to simplify the expression .

Note that the exponent doesnt become 1, but the whole expression simplifiesto be the number 1.

Example6:  Evaluate .

Note how 5 - 2 = 3, the final answers exponent.  When you multiplyyou are adding on to your exponent, so it should stand to reason that whenyou divide like bases you are taking away from your exponent.

Let's put this idea together into a general rule:

Keep in mind that you always take the numerators exponent minus yourdenominators exponent, NOT the other way around.
 

Example8:  Find the quotient .

Example10:  Simplify .

*Use def. of exponentsto evaluate

*Use def. of exponentsto evaluate

Note how 2 times 3 is 6, which is the exponent of the final answer.  We can think of this as 3 groups of 2, which of course would come out tobe 6.

Again, you can think of it as n groups ofmif it helps you to remember.
 

Example12:   Simplify .

*Use the definitionof neg. exponents to rewrite as the recip. of base

*Use the def. of exponentsto evaluate

Note how both bases of your product ended up being raised by the exponentof 3.

Example14:   Simplify .

*Mult. exponents when using powerrule for exponents

Since division is really multiplication of the reciprocal, it has thesame basic idea as when we raised a product to an exponent.

Example16:   Simplify .

*When raising a quotient to an exponent, raiseeach base of the quotient to that exponent

*Use def. of exponentsto evaluate

In other words, write it in the most condense form you can making surethat all your exponents are positive.

A lot of times you have to use more than one rule to get the job done. As long as you use the rule appropriately you should be fine. 

*When div. like basesyou subtract your exponents: -2 - (-20) = 18
 

Practice Problems

To get the most out of these, you should work the problem out onyour own and then check your answer by clicking on the link for the answer/discussionfor that  problem.  At the link you will find the answeras well as any steps that went into finding that answer.

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