Learning Objectives
Introduction
This tutorial covers the basic definition and some of the rules of exponents. The rules it covers are the product rule and quotient rule, as well as the definitions for zero and negative exponents. Exponents are everywhere in algebra and beyond. We will also dabble in looking at the basic definition of scientific notation, an application that involves writing the number using an exponent on 10. Let's see what we can do with exponents.
Tutorial
(note there are n x's
in the product)
x = base, n = exponent
The exponent tells you how many times a base appears in
a PRODUCT.
Example
1: Evaluate .
*Write the base 1/4 in a product 2 times
*Multiply
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:
Multiplying Like Bases With Exponentsin 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 3: 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 5: 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 7: Find the quotient .
Negative Exponents
Example 9: Simplify .
*Use def. of exponents to evaluate
*Use def. of exponents to evaluate
In other words, write it in the most condense form you can making sure that all your exponents are positive.
A lot of times you are having to use more than one rule to get the job done. As long as you are using the rule appropriately, you should be fine.
*Rewrite
with a pos.
exp. by taking recip. of base
*When mult.
like bases you
add your exponents
*When
div. like bases
you subtract your exponents
*Rewrite
with a pos.
exp. by taking recip. of base
A positive number is written in scientific notation if it is written in the form:
where 1 < a < 10 and r is an integer power of 10.
If the decimal point is moved to the right, the count is negative.
*Move decimal to create a number between 1 and 10
What direction did it move?
Looks like we moved it to the left.
So, our count is +8.
*Move decimal to create a number between 1 and 10
What direction did it move?
Looks like we moved it to the right.
So, our count is - 4.
Whenever you multiply by a power of 10, in essence what you are doing is moving your decimal place.
If the power on 10 is positive, you move the decimal place that many units to the right.
If the power on 10 is negative, you move the decimal place that many units to the left.
Make sure you add in any zeros that are needed
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 Problem 1a - 1d: Simplify.
Practice Problem 2a: Write the number in scientific notation.
2a. .00000146
(answer/discussion
to 2a)
Practice Problem 3a: Write the number in standard notation, without exponents.
Need Extra Help on these Topics?
http://www.sosmath.com/algebra/logs/log2/log2.html#shortcuts
This webpage helps you with the definition of exponents.
http://www.sosmath.com/algebra/logs/log3/log31/log31.html
This webpage helps with the product rule for exponents.
http://www.sosmath.com/algebra/logs/log3/log32/log32.html
This webpage helps with the quotient rule for exponents.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on July 11, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.