Learning Objectives
Introduction
Tutorial
The exponent tells you how many times a base appears in
a PRODUCT.
Example 1: Evaluate .
View a video of this example
Example 2: Evaluate .
*Negate 3 to the fourth
It may look alike, but they ARE NOT exactly the same. Can you see the difference between the two?? Hopefully, you noticed that in 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 will not get expanded like it did in example 1.
It is interpreted as finding the negative or opposite of 3 to the fourth power.
Example 3: Evaluate .
*Write the base -1/5 in a product 3 times
Specific Illustration
Note that 2 + 3 = 5, which is the exponent we ended up
with. We
had 2 x’s written in a product plus
another
3 x’s written in the product for a total
of
5 x’s in the product. To indicate
that
we put the 5 in the exponent.
Let's put this idea together into a general rule:
in general,
The reason is, exponents count how many of your base
you have in
a product. So if you are continuing that product, you are adding
on to the exponents.
Example 4: Use the product rule to simplify the expression .
Example 5: Use the product rule to simplify the expression .
Note that the exponent doesn’t become 1, but the whole expression simplifies to be the number 1.
Example 6: Evaluate .
Example 7: Evaluate .
Specific Illustration
Note how 5 - 2 = 3, the final answer’s exponent. When you multiply you are adding on to your exponent, so it should stand to reason that when you divide like bases you are taking away from your exponent.
Let's put this idea together into a general rule:
Dividing Like Bases With Exponentsin general,
Keep in mind that you always take the numerator’s
exponent minus your
denominator’s exponent, NOT the other way around.
Example 8: Find the quotient .
Example 9: Find the quotient .
Negative Exponents
Example 10: Simplify .
*Use def. of exponents to evaluate
Example 11: Simplify .
*Use def. of exponents to evaluate
Base Raised to Two Exponents
(Power Rule for Exponents)
Specific Illustration
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 to be 6.
Base Raised to two ExponentsAgain, you can think of it as n
groups of
m
if it helps you to remember.
Example 12: Simplify .
Example 13: Simplify .
*Use the definition of neg. exponents to rewrite as the recip. of base
*Use the def. of exponents to evaluate
A Product Raised to an Exponent
(Products to Powers Rule for Exponents)
Specific Illustration
Note how both bases of your product ended up being raised by the exponent of 3.
A Product Raised to an ExponentExample 14: Simplify .
Example 15: Simplify .
*Mult. exponents when using power rule for exponents
A Quotient Raised to an Exponent
(Quotients to Powers Rule for Exponents)
Specific Illustration
Since division is really multiplication of the reciprocal, it has the same basic idea as when we raised a product to an exponent.
A Quotient Raised to an ExponentExample 16: Simplify .
*When raising a quotient to an exponent, raise each base of the quotient to that exponent
*Use def. of exponents to evaluate
In other words, write it in the most condensed form you can making sure that 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.
Example 17: Simplify the exponential expression .
*When
div. like bases
you subtract your exponents: -2 - (-20) = 18
Example 18: Simplify the exponential expression .
Example 19: Simplify the exponential expression .
*Rewrite with a pos. exp. by taking recip. of base
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1f: Simplify the exponential expression.
Need Extra Help on these Topics?
http://www.sosmath.com/algebra/logs/log2/log2.html#shortcuts
This webpage gives the definition of exponents.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut24_exppart2.htm
This website helps you with some of the basic rules for exponents.
http://www.purplemath.com/modules/exponent.htm
This webpage gives an overall review of exponents.
http://www.ltcconline.net/greenl/courses/152A/polyExp/intexp.htm
This webpage goes over the rules of exponents.
http://www.sosmath.com/algebra/logs/log3/log31/log31.html
This website helps you with the product rule for exponents.
http://www.sosmath.com/algebra/logs/log3/log32/log32.html
This website helps you with the quotient rule for exponents.
http://www.sosmath.com/algebra/logs/log3/log33/log33.html
This website helps you with the rule for raising a base to two
exponents.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Feb. 15, 2008 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.