Transcript Algebra 1 - Davidsen Middle School

Algebra 1
Ch 8.2 – Zero & Negative Exponents
Objective

Students will evaluate powers that have
zero and negative exponents
Before we begin…



In the last lesson we looked at the
multiplication properties of exponents…
In this lesson we will extend and use what
we learned to include zero exponents and
negative exponents…
Let’s look at the rules…
Zero Exponents

RULE: a nonzero number raised to the
zero power is equal to 1
Example:
a0 = 1
when, a ≠ 0
Reciprocals



When working with negative exponents you need
to know what a reciprocal is…
We already covered this earlier in the course so
as a quick review…
A reciprocal is a fraction that is inverted and the
product is 1. It looks like this:
Example:
Original
2
6
Reciprocal
●
6
2
Product
=
1
Negative Exponents

Rule: a-n is the reciprocal of an
Example:
a-n
=
1
an
when, a ≠ 0
Examples

Powers with negative & zero exponents
1I
F
G
H3J
K 3
1
a
b
c
2 2
1
 2
2
(-2)0 = 1
5
x
1
 x
5
1

4
d
e
0
3
1
 3
0
Undefined – zero has no reciprocal!
Simplifying Expressions



Ok…now that you know the rules…let’s
look at simplifying some expressions…
Before we do that… be forewarned… you
need to know how to work with fractions
here!
Reminder - when multiplying fractions you
multiply the numerators and you multiply
the denominators
Example #1

Rewrite with positive exponents: 5(2-x)
When analyzing this expression I see that it has a negative exponent.
I will need to write the reciprocal of 2-x before I multiply by 5.
Don’t forget that a whole number written as a fraction is the
number over 1
Solution:
5(2-x)
5
1
 5 x  x
2
2
Example #2

Rewrite with positive exponents
2x-2y-3
When analyzing this expression I see that it has negative exponents.
I will need to write them as reciprocals before I multiply
Solution:
2x-2y-3
1
1
 2 2  3
x
y
2
 2 3
x y
Evaluating Expressions



Ok…now that you know how to simplify an
expression…Let’s look at evaluating
expressions…
You will use what you learned in this lesson
about zero and negative exponents and
combine that with what you learned about
the multiplication properties of exponents…
Again…the key is to analyze the
expression first…
Example #3
Evaluate the expression
3-2 ● 32
When analyzing this expression I see that I have a negative exponent.
But I also see that I multiplying 2 powers with the same base…
I have to make a decision here…either I work with the negative
exponent first or I work with the product of powers property…either
way I will get the same answer…
If I work with the negative exponents first….it will take me more
steps to get to the answer…so I choose to work with the
product of powers property, which states when multiplying
powers if the base is the same add the exponents…(We will look
at both solutions)
Example #3 (Continued)
Evaluate the expression
3-2 ● 32
Solution #1:
3-2 ● 32 = 3-2 + 2 = 30 = 1
Solution #2:
3-2
●
32
2
3
1
2
 2 3  2
3
3
9


1
9
Example #4
Evaluate the expression
(2-3)-2
When analyzing this expression I see that I have 2 negative
exponents. I also see that I can use the Power of a Power
Property, which states, to find the power of a power, multiply
the exponents.
Solution:
(2-3)-2 = 2-3●(-2) = 26 = 64
Simplifying Exponential Expressions



In this section we will simplify exponential
expressions, that is…we will write the
expressions with positive exponents…
Again, you will use what you learned about
zero and negative exponents and the
multiplication properties of exponents…
The key is to analyze the expression first…
Example #5
Rewrite with positive exponents
(5a)-2
When analyzing this expression I see that I can use the Power of a
Product Property, which states to find the power of a product, find
the power of each factor and multiply
Solution:
(5a)-2
=
5-2 ●
a-2
1 1
 2 2
5 a
1

25a 2
Example #6
Rewrite with positive exponents

1
d 3 n
This example is a little harder and requires some higher order
thinking skills….
First, I need to recognize that this expression is the reciprocal
of some other expression…how I recognize that is I see that it is
1 over the expression d -3n
Therefore, using the definition of a negative exponent I
can rewrite the expression as:
(d-3n)-1
Example #6 (Continued)
(d-3n)-1
Now that the expression is in a format that is not fraction form…I
see that I can use the Power of a Power Property, which states
to find the power of a power, multiply the exponents
Solution:
(d-3n)-1
= d(-3n)●(-1) = d3n
Comments

On the next couple of slides are some practice
problems…The answers are on the last slide…

Do the practice and then check your answers…If
you do not get the same answer you must
question what you did…go back and problem
solve to find the error…

If you cannot find the error bring your work to me
and I will help…
Your Turn
1.
4-2
2.
1I
F
G
H5J
K
3.
4(4-2)
4.
2-3 ● 22
5.
(-3-2)-9
1
Evaluate the exponential expression.
Write fractions in simplest form
Your Turn
Rewrite the expression with positive exponents
6.
x-5
7.
8x-2y-6
1
8.
4 x 10 y 14
9.
(-10a)0
10.
F
4 x I
G
H2 x J
K
2
1
1
Your Turn Solutions
1.
2.
3.
4.
5.
1/16
5
¼
½
-9
6.
7.
8.
9.
10.
1/x5
8/x2y6
x10/4y14
1
-1/2x3
Summary
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

A key tool in making learning effective is being able
to summarize what you learned in a lesson in your
own words…
In this lesson we talked about zero and negative
exponents. Therefore, in your own words
summarize this lesson…be sure to include key
concepts that the lesson covered as well as any
points that are still not clear to you…
I will give you credit for doing this lesson…please
see the next slide…
Credit



I will add 25 points as an assignment grade for you working on
this lesson…
To receive the full 25 points you must do the following:
 Have your name, date and period as well a lesson number as a
heading.
 Do each of the your turn problems showing all work
 Have a 1 paragraph summary of the lesson in your own words
Please be advised – I will not give any credit for work submitted:
 Without a complete heading
 Without showing work for the your turn problems
 Without a summary in your own words…