Beginning Algebra
Tutorial 29: Negative Exponents and Scientific Notation
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
Negative Exponents
or

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 . 

*Rewrite with a pos. exp. by
taking recip.
of base
*Use def. of exponents to
evaluate 
Example
2: Simplify . 

*Rewrite with a pos. exp. by
taking recip.
of base
*Use def. of exponents to
evaluate 
Simplifying an Exponential
Expression

When simplifying an exponential expression,
write it so that
each base is written one time with one POSITIVE exponent.
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. 
Except for the negative exponent rule, examples of
the following
rules can be found in Tutorial 26:
Exponents.
Product Rule:
Power Rule for Exponents:
Power of a Product:
Power of a Quotient:
Quotient Rule for Exponents:
Zero Exponent:
Negative Exponent:

Example
3: Simplify. Write answer with
positive
exponents.

Example
4: Simplify. Use positive exponents to
write
the answer.

Example
5: Simplify. Use positive exponents to
write
the answer.

Example
6: Simplify. Write answer with
positive
exponents.

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. 
Scientific Notation
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.

Writing a Number in Scientific
Notation

Step 1: Move the decimal point
so that you have
a number that is between 1 and 10. 
In other words, you will put your decimal after
the first non
zero number. 
Step 2: Count the number
of decimal places
moved in Step 1 . 
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. 
Step 3: Write as a
product of the number
(found in Step 1) and 10 raised to the power of the count (found in
Step
2). 
Example
7: Write the number in scientific
notation:
483,000,000. 

*Decimal is at the end of the
number
*Move decimal to create a
number between 1
and 10 
How many decimal places did we end up moving?
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 
Example
8: Write the number in scientific
notation:
.00054. 

*Decimal is at the beginning
of the number
*Move decimal to create a
number between 1
and 10 
How many decimal places did we end up moving?
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. 
Write a Scientific Number in
Standard Form

Basically, you just multiply the first number times
the power of
10.
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 
Example
9: Write the number in standard notation, without
exponents.


*Move the decimal 6 to the right

Example
10: Write the number in standard notation, without
exponents.


*Move the decimal 5 to the left

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.
Practice
Problem 3a: Write the number in standard
notation, without exponents.
Need Extra Help on these Topics?
Last revised on August 2, 2011 by Kim Seward.
All contents copyright (C) 2001  2010, WTAMU and Kim Seward. All rights reserved.

