Download Beginning & Intermediate Algebra, 4ed

§ 5.5
Negative Exponents and
Scientific Notation
Negative Exponents
Negative Exponents
If a is a real number other than 0 and n is an integer, then
1
n
a  n
a
Example:
Simplify by writing each expression with positive exponents.
1 1
3  2 
3
9
2
2
2x  4
x
4
Remember that without parentheses, x
is the base for the exponent –4, not 2x
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Simplifying Expressions
Example:
Simplify by writing each of the following expressions with
positive exponents.
-x
-3
1
 3
x
1
1
3   2  
3
9
2
(3) 2 
1
1

2
(3)
9
Notice the difference in results when the
parentheses are included around 3.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Simplifying Expressions
Example:
Simplify by writing each of the following expressions with
positive exponents.
1
x3
1
3

x
3 
1
x
1
3
x
2
x
y 4
1
4
2
y
x

 2
1
x
y4
(Note that to convert a power with a negative
exponent to one with a positive exponent, you
simply switch the power from a numerator to a
denominator, or vice versa, and switch the
exponent to its positive value.)
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Simplifying Expressions
Example:
Simplify by writing each of the following expressions with
positive exponents.
 3 ab 
  4 7 3 
3 a b 
2
3
2

3
3
2
4
3
ab
7
ab


3  a  b 

 3  a  b 
2
3  2
Power of a quotient rule
 2 2
4 2
3 2
7 2
2
3  2
Power of a product rule
8
8
34 a - 6b - 2
1
34 a14
1
a
= 8 - 14 6 = 8 6 2 6 = 34- 8 a14 - 6 × 2+6 = 3- 4 a8 × 8 = 4 8 = a 8
b
b 3b
3a b
3abb
81b
Power rule for exponents
Quotient and product
rules for exponents
Negative exponents
Martin-Gay, Beginning and Intermediate Algebra, 4ed
Negative exponent
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Summary of Exponent Rules
If m and n are integers and a and b are real numbers, then:
Product Rule for exponents am · an = am+n
Power Rule for exponents (am)n = amn
Power of a Product (ab)n = an · bn
Power of a Quotient

a
c
n
an
 n, c0
c
Quotient Rule for exponents
am
mn

a
, a0
n
a
Zero exponent a0 = 1, a  0
Negative exponent
a n 
1
, a0
n
a
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Scientific Notation
In many fields of science we encounter very large or
very small numbers. Scientific notation is a
convenient shorthand for expressing these types of
numbers.
A positive number is written in scientific notation if
it is written as a product of a number a, where
1  a < 10, and an integer power r of 10.
a  10r
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Scientific Notation
To Write a Number in Scientific Notation
1) Move the decimal point in the original number to the
left or right so that the new number has a value between
1 and 10.
2) Count the number of decimal places the decimal point is
moved in Step 1. If the original number is 10 or greater,
the count is positive. If the original number is less than
1, the count is negative.
3) Multiply the new number in Step 1 by 10 raised to an
exponent equal to the count found in Step 2.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Scientific Notation
Example:
Write each of the following in scientific notation.
Have to move the decimal 3 places to the left, so that the
1) 4700
new number has a value between 1 and 10.
Since we moved the decimal 3 places, and the original
number was > 10, our count is positive 3.
4700 = 4.7  103
2)
0.00047
Have to move the decimal 4 places to the right, so that
the new number has a value between 1 and 10.
Since we moved the decimal 4 places, and the original
number was < 1, our count is negative 4.
0.00047 = 4.7  10-4
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Scientific Notation
To Write a Scientific Notation Number in
Standard Form
•
Move the decimal point the same number of
spaces as the exponent on 10.
• If the exponent is positive, move the
decimal point to the right.
• If the exponent is negative, move the
decimal point to the left.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Scientific Notation
Example:
Write each of the following in standard notation.
1) 5.2738  103
Since the exponent is a positive 3, we move the decimal 3
places to the right.
5.2738  103 = 5273.8
2)
6.45  10-5
Since the exponent is a negative 5, we move the decimal
5 places to the left.
00006.45  10-5 = 0.0000645
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Operations with Scientific Notation
Multiplying and dividing with numbers written in scientific
notation involves using properties of exponents.
Example:
Perform the following operations.
1) (7.3  10-2)(8.1  105) = (7.3 · 8.1)  (10-2 · 105)
= 59.13  103
= 59,130
1.2 10 4 1.2 10 4
5

0
.
3

10


 0.000003
2)
9
9
4 10
4 10
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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