Transcript Section R.2
R.2
Integer Exponents, Scientific
Notation, and Order of Operations
Simplify expressions with integer
exponents.
Solve problems using scientific
notation.
Use the rules for order of operations.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Integers as Exponents
When a positive integer is used as an exponent, it indicates
the number of times a factor appears in a product.
For any positive integer n,
a
n
a a a a ... a ,
b a se
n fa cto rs
where a is the base and n is the exponent.
Example: 84 = 8 • 8 • 8 • 8
For any nonzero real number a and any integer m,
a0 = 1 and a m a1 .
m
Example: a) 80 = 1
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
x
b)
y
2
5
x
2
1
y
5
1
x
2
y
5
y
5
x
2
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Properties of Exponents
Product rule
a a a
m
n
Raising a product to a power
mn
(ab)m = ambm
Quotient rule
a
a
Raising a quotient to a power
m
n
a
mn
( a 0)
a
b
m
a
m
b
m
( b 0)
Power rule
(am)n = amn
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Examples – Simplify.
a) r 2 • r 5
= r (2 + 5) = r 3
b)
36 y
18 y
9
4
36
y
94
d) (3a3)4 = 34(a3)4
= 81a12 or
e)
18
2y
5
2
2 1a b
4
7
c
2
3
3a 2 b 2
4
c
3
c) (p6)4 = p ‒24 or
6
3 a b
c
1
p
24
6
27 c
12
a
12
3
6
6
6
27 b c
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a
12
6
a b
81
12
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Scientific Notation
Use scientific notation to name very large and very
small positive numbers and to perform
computations.
Scientific notation for a number is an expression of
the type N 10m,
where 1 N < 10, N is in decimal notation, and m is
an integer.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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Examples
Convert to scientific notation.
a) 17,432,000 = 1.7432 107
b) 0.00000000024 = 2.4 1010
Convert to decimal notation.
a) 3.481 106 = 3,481,000
b) 5.874 105 = 0.00005874
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Another Example
Chesapeake Bay Bridge-Tunnel. The 17.6-mile-long
tunnel was completed in 1964. Construction costs
were $210 million. Find the average cost per mile.
2.1 10
8
1.76 10
1
1.19 10
8 1
$1.19 10
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
7
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Rules for Order of Operations
1. Do all calculations within grouping symbols before
operations outside. When nested grouping symbols
are present, work from the inside out.
2. Evaluate all exponential expressions.
3. Do all multiplications and divisions in order from
left to right.
4. Do all additions and subtractions in order from left
to right.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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Examples
a) 4(9 6)3 18 = 4(3)3 18
= 4(27) 18
= 108 18
= 90
b)
15 (7 2 ) 20
3 2
3
2
15 5 20
27 4
3 20
31
23
31
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