Download R.2 Integer Exponents, Scientific Notation, and Order of Operations

CHAPTER R:
Basic Concepts of Algebra
R.1 The Real-Number System
R.2 Integer Exponents, Scientific Notation, and Order of
Operations
R.3 Addition, Subtraction, and Multiplication of Polynomials
R.4 Factoring
R.5 The Basics of Equation Solving
R.6 Rational Expressions
R.7 Radical Notation and Rational Exponents
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
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 ,
base
n factors
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
b)
x 2
1
1
y5
2
5
 x  5  2  y  2
y 5
y
x
x
Slide R.2 - 4
Properties of Exponents
Product rule
a a  a
m
n
Raising a product to a power
mn
(ab)m = ambm
Quotient rule
am
mn

a
an
Raising a quotient to a power
(a  0)
m
am
a
   m
b
b
(b  0)
Power rule
(am)n = amn
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.2 - 5
Examples – Simplify.
a) r 2 • r 5
= r (2 + 5) = r 3
d)
(3a3)4 = 34(a3)4
81
= 81a12 or 12
a
b) 36 y 9 36
94
5

y

2
y
18 y 4 18
e)
c) (p6)4 = p ‒24 or 124
p
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
3
3
 21a b 
 3a b 

 7c 4 
 c 4 




33 a 6b 6 a 6b 6


12
c
27c12
a6

27b6 c12
2 2
2 2
Slide R.2 - 6
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
Slide R.2 - 7
Examples
Convert to scientific notation.
a) 17,432,000 = 1.7432  107
b) 0.00000000024 = 2.4  1010
Convert to decimal notation.
a) 3.481  106 = 3,481,000
b) 5.874  105 = 0.00005874
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.2 - 8
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 108
81
 1.19  10
1
1.76  10
 $1.19  107
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.2 - 9
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
Slide R.2 - 10
Examples
a) 4(9  6)3  18 = 4(3)3  18
= 4(27)  18
= 108  18
= 90
b) 15  (73  2)2  20  15  5  20
3 2
27  4
3  20 23


31
31
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.2 - 11