**Learning Objectives**

After completing this tutorial, you should be able to:

- Simplify exponential expressions involving negative exponents.
- Write a number in scientific notation.
- Write a number in standard notation, without exponents.

** Introduction**

This tutorial picks up where **Tutorial
26: Exponents** left off. It finishes the rules of
exponents
with negative exponents. Also we will go over scientific
notation.
Like
it or not, the best way to master these exponents is to work through
exponent
problems. So I guess we better get to it.

** Tutorial**

Be careful with negative
exponents. The
temptation is to negate the base, which would not be a correct thing to
do. **Since exponents
are another
way to write multiplication and the negative is in the exponent, to
write
it as a positive exponent we do the multiplicative inverse which is to
take the reciprocal of the base.**

** Example
1:** 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.

**Product Rule: **

**Power Rule for Exponents: **

**Power of a Product: **

**Power of a Quotient: **

**Quotient Rule for Exponents: **

**Zero Exponent: **

**Negative Exponent: **

***Rewrite
with a pos.
exp. by taking recip. of base **

***Rewrite
with a pos.
exp. by taking recip. of base**

***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 **

Be careful going into the last line. Note that
you do not see
an exponent written with the number 5. This means that the
exponent
on 5 is understood to be 1. Since it doesn't have a
negative
exponent, we DO NOT take the reciprocal of 5. The only base that
has a negative exponent is *a*, so *a* is the only base we take the reciprocal of.

**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.**

In other words, you will put your decimal after the first non zero number.

If the decimal point was moved to the left, the count
is positive.

If the decimal point is moved to the right, the count is negative.

***Move decimal to create a
number between 1
and 10**

We started at the end of the number 483000000 and moved it between the 4 and 8. That looks like a move of 8 places.

**What direction did it move?**

Looks like we moved it to the left.

**So, our count is +8.**

Note how the number we started with is a bigger number
than the one
we are multiplying by in the scientific notation. When that is
the
case, we will end up with a positive exponent

***Move decimal to create a
number between 1
and 10**

We started at the beginning of the number .00054 moved it between the 5 and 4. That looks like a move of 4 places.

**What direction did it move?**

Looks like we moved it to the right.

**So, our count is - 4.**

Note how the number we started with is a smaller number
than the one
we are multiplying by in the scientific notation. When that is
the
case we will end up with a negative exponent.

**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**

These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. **Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

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 - 1b:Simplify, use positive exponents to write each answer.

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?**

**The following are webpages
that can assist
you in the topics that were covered on this page: **

**http://www.sosmath.com/algebra/logs/log3/log32/log32.html**

This webpage helps with the quotient rule for exponents.

**http://www.purplemath.com/modules/exponent.htm**

This webpage gives an overall review of exponents. It contains
rules from both this tutorial and Tutorial 26: 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 August 2, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.