Intermediate Algebra
Tutorial 23: Exponents and Scientific Notation, Part I

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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
Definition of Exponents
(note there are n x's
in the product)
x = base, n =
exponent

Exponents are
another way to
write multiplication.
The exponent tells you how many times a base appears in
a PRODUCT.
Example
1: Evaluate .


*Write the base 2 in a
product 3 times
*Multiply 
Example
2: Evaluate . 

*Write the base 1/4 in a
product 2 times
*Multiply

Multiplying Like Bases With
Exponents
(The Product Rule for Exponents)
Specific Illustration

Let’s first start by using the definition
of exponents to help you to understand how we get to the law for multiplying like
bases
with exponents:
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
Exponents
(The Product Rule for Exponents)
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 . 

*When mult. like bases you add
your exponents 
Example
4: Use the product rule to simplify the
expression . 

*When mult. like bases you add
your exponents

Zero as an Exponent

Except for 0, any base raised
to the 0 power
simplifies to be the number 1.
Note that the exponent doesn’t become 1, but the whole
expression simplifies
to be the number 1.
Example
5: Evaluate . 

*Any expression raised to the 0
power simplifies
to be 1

Example
6: 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. 

*x raised
to the
0 power is 1
*Multiply

Dividing Like Bases With
Exponents
(Quotient Rule for Exponents)
Specific Illustration

Let’s first start by using the definition
of exponents to help you to understand how we get to the law for dividing like bases
with exponents:
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
Exponents
(Quotient Rule for Exponents)
in general,

In other words, when you
divide like bases
you subtract their exponents.
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 . 

*When div. like bases you subtract
your exponents

Example
8: Find the quotient . 

*When div. like bases you
subtract your exponents

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
9: Simplify . 

*Rewrite with a pos. exp. by
taking recip.
of base
*Use def. of exponents to
evaluate 
Example
10: 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. 
Example
11: Simplify. Write answer with
positive
exponents.

Example
12: Simplify. Write answer with
positive
exponents.

Example
13: 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
14: 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
15: 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
16: Write the number in standard notation, without
exponents.


*Move the decimal 6 to the right

Example
17: 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
Problem 1a  1d: Simplify.
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 July 11, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.
