**Learning Objectives**

After completing this tutorial, you should be able to:

- Simplify exponential expressions involving raising a base to two exponents, raising a product to an exponent and raising a quotient to an exponent.
- Multiply and divide numbers written in scientific notation.

** Introduction**

This tutorial picks up where **Tutorial
23: Exponents and Scientific Notation Part I** left off. It
finishes the rules of exponents with raising a base to two exponents,
raising
a product to an exponent and raising a quotient to an exponent.
Also
we will revisit the concept of 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**

Let’s first start by using the **definition
of exponents** as well as the **law
for multiplying like bases** (both found in **Tutorial
23: Exponents and Scientific Notation Part I**, to help you to
understand
how we get to the law for raising a base to two exponents:

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.

In other words, **when you
raise a base to two
exponents, you multiply those exponents together.**

Again, you can think of it as *n* groups of *m* if it helps you to remember.

** Example
1: **Simplify .

Let’s first start by using the **definition
of exponents, found in Tutorial 23: Exponents and Scientific Notation
Part
I**, to help you to understand how we get to the law for raising
a product to an exponent:

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

In other words, **when you have
a PRODUCT (not
a sum or difference) raised to an exponent, you can simplify by raising
each base in the product to that exponent.**

** Example
2: **Simplify .

Let’s first start by using the **definition
of exponents, found in Tutorial 23: Exponents and Scientific Notation
Part
I**, to help you to understand how we get to the law for raising
a quotient to an exponent:

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

In other words, **when you have
a QUOTIENT (not
a sum or difference) raised to an exponent, you can simplify by raising
each base in the numerator and denominator of the quotient to that
exponent.**

** Example
3: **Simplify .

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

***Use def. of exponents to
evaluate**

Of course, we all know that life isn’t so cut and
dry. A lot
of times you are needing to use more than one definition or law of
exponents
to get the job done. What we did above and in **Tutorial
23: Exponents and Scientific Notation Part I** was to set the
foundation
to make sure you have a good understanding of the different ideas
associated
with exponents. Next we will work through some problems which
will
intermix these different laws.

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

***Raise each
base to 10**

***Mult.
your exponents**

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

Not quite finished. Remember that the number has
to be between
1 and 10 for it to be in scientific notation.

***When
div. like bases you sub. your exp. **

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

Practice Problems 2a - 2b:Perform indicated operations, write each result in scientific notation.

** Need Extra Help on these Topics?**

**http://www.sosmath.com/algebra/logs/log3/log33/log33.html**

This webpage helps with the rule for raising a base to two
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 23: Exponents and Scientific
Notation Part I.

**http://www.ltcconline.net/greenl/courses/152A/polyExp/intexp.htm**

This webpage gives an overall review of exponents. It contains
rules from both this tutorial and Tutorial 23: Exponents and Scientific
Notation Part I.

**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 12, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.