Lesson 8.2 - Negative and Zero Exponents.ppt

A negative exponent represents the reciprocal of the base raised to the positive version of the exponent. For example,
𝑎
−
𝑛
=
1
𝑎
𝑛
a
−n
=
a
n

1
​
. This means if you have a base with a negative exponent, you can flip it into the denominator and make the exponent positive. This rule simplifies the handling of expressions with negative exponents.

1. 8.2 Negative and Zero
Exponents
I love exponents!

2. 4 3 2 1 0
In addition to level 3,
students make
connections to other
content areas and/or
contextual situations
outside of math.
 
Students will construct,
compare, and interpret linear
and exponential function
models and solve problems in
context with each model.
- Compare properties of 2
functions in different ways
(algebraically, graphically,
numerically in tables, verbal
descriptions)
- Describe whether a
contextual situation has a linear
pattern of change or an
exponential pattern of change.
Write an equation to model it.
- Prove that linear functions
change at the same rate over
time.
- Prove that exponential
functions change by equal
factors over time.
- Describe growth or decay
situations.
- Use properties of exponents
to simplify expressions.
Students will
construct,
compare, and
interpret linear
function models
and solve
problems in
context with the
model.
- Describe a
situation where
one quantity
changes at a
constant rate per
unit interval as
compared to
another.

 
Students will have
partial success at
a 2 or 3, with
help.
Even with help,
the student is
not successful at
the learning
goal.
Focus 8 Learning Goal – (HS.N-RN.A.1 & 2, HS.A-SSE.B.3, HS.A-CED.A.2, HS.F-IF.B.4, HS.F-
IF.C.8 & 9, and HS.F-LE.A.1) = Students will construct, compare and interpret
linear and exponential function models and solve problems in context
with each model.

3. Definition of Negative Exponents
(let a be a nonzero number and let n be a positive integer)
The expression a
-n
is the reciprocal of a
n
.
a
-n
= 1 a ≠ 0
a
n
1 = a
n
a
-n
3
-2
= 1
3
2
= 1
9

4. The negative exponent says the number
needs to be moved to the opposite location
and made positive.
If it’s negative in the numerator, it
belongs in the denominator position
positive.
If it’s negative in the denominator
position, it belongs in the numerator
position positive.

5. Definition of Zero Exponent
(let a be a nonzero number and let n be a positive integer)
A nonzero number to the zero power is
ALWAYS 1!
a
0
= 1a  0
3
0
= 1
(x
2
y
5
)
0
= 1
The expression 0
0
is undefined.

6. Simplify expressions: write with positive
exponents.
(-5)
-3
=
244
-3
=

-3
-4
=

1
(-5)
3
= - 1
125
24 ∙ 1
4
3
= 24∙ 1 = 3
64 8
1
-3
4
1


81
= -

7. Simplify expressions: write with positive
exponents.
3a
-3
b
-2
=
3
a
3
b
2
(3
-3
)
2
=

3
(-3 ∙ 2)
= 3
-6
=

1
3
6
=
1
729

8. Graphing with a variable
as an exponent
x321 0 -1 -2 -3
2
x
8422
0
=12
-1
= ½ 2
-2
= ¼ 2
-3
=
1
/
8
Sketch the graph of y = 2
x
Will it ever touch the x-axis?

9. Example: (Just follow this example to see what you get to do on
your assignment.)
Between 1970 and 1990, the Missouri
population increased at a rate of .47%
per year. The population P in t years is
given by:
P = 4,903,000 1.0047
t
Where t = 0 for 1980

10. To find the population, plug the numbers into
the formula and then use a calculator.
Find the population in 1970, 1980, and 1990
Pop in 1970t = -10
Pop in 1990t = 10
P = 4,903,000  1.0047
-10
(set up the problem)
= 4,678,406 (calculate)
This is the population in 1970.
Do you expect it to be more
or less in 1970 than 1980?

11. Population in 1990:
P = 4,903,000  1.0047
10
P = 5,138,376