to investigate the meaning of non-positive exponents. to write a number with a negative exponent in...
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• 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?
In this lesson you will learn what a zero or a negative integer means as an exponent.
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
xx
8zz
5
22
3
3
22
5
5
xx
6
3
mm
3
5
55
2y 23 07 42 3x
7z 02 0x 3m 25
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?
7
5
yy
2
4
33
4
4
77
3
6
xx
8zz
5
22
3
3
22
5
5
xx
6
3
mm
3
5
55
2y 23 07 42 3x
7z 02 0x 3m 25
•Step 3▫Go back to the expressions in Step 1 that
resulted in a negative exponent. Write each in expanded form. Then reduce them.
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 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.
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 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.
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 A
•Use the properties of exponents to rewrite each expression without a fraction bar.
5
7
34 8
25x
3
8
52
8
8
3(17)17
55 5 7
7 7
3 13 3 4
4 4 8
8
125 25 x
x
3 3 88
15 5 2
2
03 17 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?
▫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 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.
▫ 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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