Transcript Lesson 6.6: Zero and Negative Exponents

•
•
•
To investigate the meaning of non-positive exponents.
To write a number with a negative exponent in a form that
has a positive exponent and write a number with a positive
exponent in a form that has a negative exponent.
To write in scientific notation numbers close to zero.
Have you noticed that so far in this chapter
the exponents have been positive integers?
x
0
x
2
In this lesson you will learn what a zero or
a negative integer means as an exponent.
More Exponents
• Step 1
▫ Use the division property of exponents to rewrite
each of these expression with a single exponent.
y7
2

y
y5
32
2
4  3
3
74
0
4  7
7
2
4
5  2
2
x3
3
6  x
x
z8
 z7
z
23
0

2
23
x5
0

x
x5
m6
3

m
m3
53
2

5
55
Some of your answers in Step 1 should have been
positive exponents, some should have been negative,
and some should have been zero exponents.
• Step 2
▫ How can you tell what type of exponent will result
simply by looking at the original expression?
y7
2

y
y5
32
2
4  3
3
74
0
4  7
7
2
4
5  2
2
x3
3
6  x
x
z8
 z7
z
23
0

2
23
x5
0

x
x5
m6
3

m
m3
53
2

5
55
• Step 3
▫ Go back to the expressions in Step 1 that resulted
in a negative exponent. Write each in expanded
form. Then reduce them.
32
2

3
34
33
3333
1
1
 2 
9
3
3
2
x
4
3
5  2

x
2
x6
2
xxx
22222 x  x  x  x  x  x
1
1
1
 4 
 3
16
2
x
53
2

5
55
555
55555
1
1
 2 
25
5
• Step 4
▫ Compare your answer from step 3 and Step 1. Tell
what a base raised to a negative exponent means.
• Step 5
▫ Go back to the expressions in Step 1 that resulted
in an exponent of zero. Write each in expanded
form. Then reduce them.
23 2  2  2

1
3
222
2
74 7  7  7  7
x5  x  x  x  x  x  1

1 5
4
xxxxx
7777
x
7
• Step 6
▫ Compare your answers from Step 5 and Step 1.
Tell what a base raised to an exponent of zero
means.
• Step 7
▫ Use what you have learned about negative
exponents to rewrite each of these expressions
with positive exponents and only one fraction bar.
1
52
 2
5
1
1
1
 8
8
3
3
4x 2
4y 5
2 5 
z y
x2 z2
• Step 8
▫ In one or two sentences, explain how to rewrite a
fraction with a negative exponent in the
numerator or denominator as a fraction with
positive exponents.
3
Exponential Form
Fraction Form
33
27
32
9
31
3
30
1
3-1
1/3
3-2
1/9
3-3
1/27
For any nonzero value of b and for any
value of n,
b
n
1
 n
b
1
n
b
n
b
0
b 1
Example A
• Use the properties of exponents to rewrite each
expression without a fraction bar.
35
35
1
5
5
7


3


3

4
47
47
47
1
53
3
3
8

5


5

2
28
28
1
25
8

25


25

x
x8
x8
3(17)8
178
 3  170  3
Example B
• Solomon bought a used car for $5,600. He
estimates that it has been decreasing in value by
15% each year.
▫ If his estimate of the rate of depreciation is
correct, how much was the car worth 3 years ago?
x
3
y

A
(1

r
)

5600(1

0.15)
$9,118.66
▫ If the car is 7 years old, what was the
original price
of the car?
5600(1  0.15)7  $17, 468.50
Example C
• Convert each number to standard notation from
scientific notation, or vice versa.
▫ A pi meson, an unstable particle released in a
nuclear reaction, “lives” only 0.000000026
seconds.
▫ The number 6.67 x 10-11 is the gravitational
constant in the metric system used to calculate the
gravitational attraction between two objects that
have given masses and are a given distance apart.
▫ The mass of an electron is 9.1 x 10-31 kg.
Example C
• Convert each number to standard notation from scientific notation, or vice
versa.
▫ A pi meson, an unstable particle released in a nuclear reaction, “lives”
only 0.000000026 seconds.
0.000000026 
2.6
2.6
8


2.6
x10
100,000,000 108
▫ The number 6.67 x 10-11 is the gravitational constant in the metric system
used to calculate the gravitational attraction between two objects that
have given masses and are a given distance apart.
6.67
6.67 x 10 
 0.0000000000667
The mass of an electron10
is 11
9.1 x 10-31 kg.
11
▫
9.1 x 1031 
9.1
 0.00000000000000000000000000000091
31
10