Transcript sec_5_2 - Benjamin N. Cardozo High School

Section 5.2
Negative
Exponents and
Scientific
Notation
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Negative Exponents
Definition of a Negative Exponent
x
n
1
 n
x
where x  0
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Example
Write with positive exponents.
a. h
b.

4
2a
 14
h

3 5

1
 2a 
3
5

1
32a15
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Negative Exponents
Laws of Exponents Where x, y, ≠ 0
The Product Rule
x a  x b  x ab
The Quotient Rule
a
x
Use if a > b.
a b

x
xb
xa
1
 b a
b
x
x
Power Rules
 xy 
a
 xay a,
x 
a
Use if a < b.
a
b
 x ab ,
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x
xa
   a
y
y
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Negative Exponents
Properties of Negative Exponents Where x, y, ≠ 0
1
n

x
x n
x m y n
 m
n
y
x
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Example
Simplify. Write the expression with no negative exponents.
3
5
3 2
5
a.
x y x x  x
5
6
2
yy
y
x y
b.


4 2 3
2ab c
 23 a3b( 4)( 3) c(2)( 3)
3
3 12 6
2 a b c
12
b
 3 3 6
2 a c
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Scientific Notation
Scientific Notation
A positive number is written in scientific notation if it
is in the form a × 10n, where 1  a  10 and n is an
integer.
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Scientific Notation
8200 =
 1000 = 8.2  103
8.2
Greater than 1 and
less than 10
Power
of 10
34,200,000 = 3.42  10000000 = 3.42  107
Scientific notation
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Example
Write 67,300 in scientific notation.
What power?
67,300. = 6.73  10n
Starting position
of decimal point
Ending position
of decimal point
The decimal point was moved 4 places to the left, so
we use a power of 4.
67,300 = 6.73  104
A number that is larger than 10 and written in scientific notation will
always have a positive exponent as the power of 10.
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Example
Write 0.048 in scientific notation.
What power?
0.048 = 4.8  10n
Starting position
of decimal point
Ending position
of decimal point
The decimal point was moved 2 places to the right, so
we use a power of –2.
0.048 = 4.8  10–2
A number that is smaller than 1 and written in scientific notation
will always have a negative exponent as the power of 10.
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Example
Write 9.1  104 in decimal notation.
9.1  104 = 9.1000  104 = 91,000
Move the decimal point
4 places to the right.
Write 6.72  10–3 in decimal notation.
6.72  10–3 =
6.72  10–3 = 0.00672
Move the decimal point
3 places to the left.
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Example
Use scientific notation and the laws of exponents to find
the following. Leave your answer in scientific notation.
32,000,0001,500,000,000,000


 3.2  107 1.5  1012

Write each number in scientific
notation.
 3.2  1.5  107  1012
Rearrange the order.
 4.8  1019
Multiply.
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Example
Use scientific notation and the laws of exponents to find
the following. Leave your answer in scientific notation.
6.3  104
0.00063

2.1 102
0.021
Write each number in scientific
notation.
6.3 104

 2
2.1 10
Rearrange the order.
6.3 102

 4
2.1 10
Rewrite with positive exponents.
 3.0  102
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