our goal: to write a number with a negative exponent in a form that has a positive exponent and...
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• OUR GOAL: 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.
Lesson 4.6: Zero and Negative Exponents
In this lesson you will learn what a zero or a negative integer means as an exponent. You will then learn how to manipulate them so you always have positive exponents as your answer.
0x 2x
More Exponents
Step 1 Use the division property of exponents to rewrite
each of these expression with a single exponent.
7
5
yy
2
4
33
4
4
77
3
6
xx5
22
2y 23 07 42 3x
Notice that some of your answers in Step 1 are positive exponents, some are negative, and some are
zero exponents.
Step 2 Go back to the expressions in Step 1 that resulted in
a negative exponent. Write each in expanded form (using the cross out method). Then write the answer as a fraction.
2
4
33
3
6
xx5
22
3
5
55
23 42 3x 25
2
3 33 3 3 3
1 193
4
22 2 2 2 2
1 1162
3
1
x x xx x x x x x
x
2
5 5 55 5 5 5 5
1 1255
Step 3 Compare your answer from step 2 and Step 1. Tell
what a base raised to a negative exponent means.
Step 4 Go back to the expressions in Step 1 that resulted in
an exponent of zero. Write each in expanded form. Then reduce them.
4
4
77
3
3
22
5
5
xx
2 2 21
2 2 2
7 7 7 71
7 7 7 7
1x x x x xx x x x x
Step 5 Compare your answers from Step 4 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.
251
8
13
2
2 5
4xz y
2
15
8
13
5
2 2
4yx z
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.
Exponential Form Fraction Form
33 27
32 9
31 3
30 1
3-1 1/3
3-2 1/9
3-3 1/27
3
For any nonzero value of b and for any value of n,
1nn
bb
1 nn
bb
0 1b
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?
If the car is 7 years old, what was the original price of the car?
3(1 ) 5600(1 0.15) $9,118.66xy A r
75600(1 0.15) $17, 468.50
Example CConvert 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.
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.
88
2.6 2.60.000000026 2.6 x10
100,000,000 10
1111
6.676.67 10 0.0000000000667
10x
3131
9.19.1 10 0.00000000000000000000000000000091
10x
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