Download 1-4 zero and negative exponent investigation

Math 2
Lesson 1-4: Zero & Negative Exponents
Name ______________________________
Date ____________________________
Learning Goal:

I can use exponent rules to simplify expressions with exponents of zero and negative
exponents.
The pond at North Chagrin Reservation fills up with lily pads every year. Lily pads reproduce
rapidly, doubling in population every day! You go the pond today (day zero) and count 32 lily
pads. A week later, you return to find 4096 lily pads!
1.
Fill in the following table.
Day
Number of Lily
Pads
Value written as a
power of 2.
-4
four days ago
-3
-2
-1
0
today
1
tomorrow
2
32
64
32  21
32  2 2
3
4
5
6
7
one week from
today
4096
32  2 7
2.
What happened to the lily pad population as time moved forward?
3.
What happened to the lily pad population as time moved backward?
4.
What happened when your exponent was 0?
5.
What happened when your exponent was negative?
Your parents bought you a used car today that is five years old & fully loaded! You figure you’ll
keep the car for about four years and then sell it for something newer in college. You find out
that cars typically depreciate by 20% every year, meaning that they lose value as time goes on.
That also means that they retain 80% of their value each year. You find out that your parents paid
$10,000 for the car.
6.
Fill in the following table.
Year
Value of Car
Value written as
a power of 0.80
-5
year the car was
made
-4
-3
-2
-1
0
today
1
$10,000
10,000(0.80)1
2
3
4
year you think
you’ll sell it
10,000(0.80)4
7.
What happened to the value of the car as time moved forward?
8.
What happened to the value of the car as time moved backward?
9.
What happened when your exponent was 0?
10.
What happened when your exponent was negative?
11.
Filling out the tables on the previous two pages led to some pretty interesting properties of
exponents. First, let’s explore what happens when we have an exponent of zero.
32  20  _____
12.
10,000(0.80)0  ____________
20  ______
(0.80)0  ______
Let’s see what happens when we raise other numbers (bases) to the power of 0.
50  _____
61160  _____
x 0  _____
 2 
0
 _____
 89 
0
 _____
 0  _____
13.
ZERO POWER PROPERTY:
When b is any base and b  0 , b0  _____
14.
Evaluate the following.
21  _____
15.
16.
23  _____
24  _____
Now let’s explore what happens when we have a negative exponent.
32  21  _____
32  22  _____
32  23  _____
32  24  _____
1
32     _____
2
1
32     _____
4
1
32     _____
8
 1
32     _____
 16 
So what can we conclude from the above results? Answer the following using fractions
only!
21  _____
17.
22  _____
22  _____
23  _____
24  _____
Let’s see what happens when we raise other numbers (bases) to negative exponents. Write
your answers as simplified fractions.
31  _____
82  _____
43  _____
53  _____
18.
Let’s convert our answers from above to fractions with positive exponents. Fill in the
blanks.
1
31 
3
19.
1
82 
1
43 
8
53 
1
5
4
One conclusion we can draw about negative exponents is…
NEGATIVE EXPONENTS PROPERTY:
When b is any base and b  0 ,
b x 
1
b
In the used car problem, we multiplied the value of the car by 0.80, because even though the car
lost 20% of its value, it retains 80% of its value each year. 80% can also be written as the
4
fraction .
5
20.
Simplify the following using properties discovered in Lesson 1-3. Write your answers as
simplified fractions.
1
4
  
 5
2
4
  
 5
______________
3
______________
4
  
 5
4
______________
4
  
 5
______________
In the used car problem, when we went back in time, the value of the car grew, but not by 20%.
The value of the car grew by a factor of 1.25 (25% growth rate).† 1.25 can also be written as the
5
.
4
fraction
† Ask your teacher if you’re curious why this is.
21.
Simplify the following. Write your answers as simplified fractions.
1
4
  
 5
22.
______________
3
______________
4
  
 5
4
______________
4
  
5
______________
Let’s convert our answers from above to fractions with positive exponents. Fill in the
blanks.
1
5
4

 
5
4
23.
2
4
  
5
2
5
4

 
5
4
3
5
4

 
5
4
The other conclusion we can draw about negative exponents is…
4
5
4

 
5
4
NEGATIVE EXPONENTS PROPERTY:
When a and b are not equal to zero,
24.
a
 
b
x

b
a
Simplify the following. Use only positive exponents in your answers.
1
x
 y 
 
2
2
______________
a
  
b
______________
 5x 
 7y  
 
3
 2h 
 j  
 
______________
______________
PRACTICE:
Simplify completely. Leave your answers as fractions. No negative exponents!
1.)
5 3 
2.)
7 2 
3.)
6 1 
4.)
b 7 
5.)
2
  
5
6.)
 3
  
2
7.)
a
 
b
8.)
 x 
 2y  
 
9.)
19 0 
10.)
 7 
11.)
y0 
12.)  abcdefg  
4
1
3
8
0

0
What if a negative exponent appears in the bottom of a fraction?
1
23

2 3 1
1
x5

x 5 1
4
b 4 c 4
b


 
c 4 b4
c
Simply put, if you get a base with a negative exponent in the denominator, move it to
the numerator with a positive exponent.
Simplify completely. Leave your answers as fractions. No negative exponents!
13.)
1

y 6
14.)
1

8 3
15.)
x 7

y 5
16.)
23 w6 x 4

y 5 z 2