r.2 integer exponents, scientific notation, and order of operations simplify expressions with...
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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
Slide R.2 - 2Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Integers as ExponentsWhen 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,
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 .
Example: a) 80 = 1 b)
factors
... ,n
base n
a a a a a a
1 m
m
aa
2 52 5
5 5 2 2
1 1x yx y
y y x x
Slide R.2 - 3Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Properties of Exponents
Product rule
Quotient rule
Power rule
(am)n = amn
Raising a product to a power
(ab)m = ambm
Raising a quotient to a power
m n m na a a
( 0)m
m nn
aa a
a
( 0)m m
m
a ab
b b
Slide R.2 - 4Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples – Simplify.
a) r 2 • r 5
= r (2 + 5) = r 3
b)
c) (p6)4 = p ‒24 or
d) (3a3)4 = 34(a3)4
= 81a12 or
e) 9
4
36
18
y
y2 32
4
21
7
a b
c
24
1
p
12
81
a
2 2
4
33a b
c
3 6 6
12
3 a b
c
6 6
1227
a b
c
6
6 1227
a
b c
9 436
18y
52y
Slide R.2 - 5Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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.
Slide R.2 - 6Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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
Slide R.2 - 7Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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.
88 1
1
7
2.1 101.19 10
1.76 10
$1.19 10
Slide R.2 - 8Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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.
Slide R.2 - 9Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Examples
a) 4(9 6)3 18 = 4(3)3 18
= 4(27) 18
= 108 18
= 90
b) 3 2
15 (7 2) 20 15 5 20
3 2 27 43 20 23
31 31