**Learning Objectives**

After completing this tutorial, you should be able to:

- Use the definition of exponents.
- Simplify exponential expressions involving multiplying like bases, zero as an exponent, dividing like bases, and negative exponents.
- Write a number in scientific notation.
- Write a number in standard notation, without exponents.

** 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:**

*in general,*

In other words, **when you
multiply like bases
you add your exponents**.

**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 .

Be careful on this example. As shown in **Tutorial ****4: Operations on Real Numbers****, **the order of operations says to
evaluate exponents before doing any multiplication. This means we
need to find *x *raised to the 0 power
first
and then multiply it by 3.

**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:**

*in 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 .

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
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 **

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 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.