Transcript Can we put a picture of my book cover here?

Chapter 2
Variables and Exponents
Section 2.1
Simplifying Expressions
What are like terms?
Like terms contain the same variables
with the same exponents.
Example 1
1 2
p like terms?
a) Are 2 p ,  0.1 p , and
5
2
2
Yes. Each variable contains the variable p with
2
and exponent of 2. They are p -terms.
b) Are
18 f 3 ,  f 4 , and 0.4 f like terms?
No. Although each contains the variable f, the
exponents are not the same.
Combine like terms
We combine like terms using the
distributive property. We can add
and subtract only those terms that
are like terms.
Example 2
Combine like terms.
a)
3d  8  5d  1  (3  5)d  (8  1)
 2d  7
We combine the coefficients of the like terms.
b) 9w  4w  w  3w  9w  w  4w  3w
3
3
3
3
 8w  7 w
3
Parentheses in an Expression
If an expression contains parentheses, use
the distributive property to clear the
parentheses, and then combine like terms.
Example 3
Combine like terms and simplify.
5(3n  2)  (4n  9)  5(3n  2)  1(4n  9)
 15n  10  4n  9
 15n  4n  10  9
 19n  1
Translate English Expressions
to Mathematical Expressions
Read the phrase carefully, choose a variable
to represent the unknown quantity, then
translate the phrase to a mathematical
expression.
Example 4
Write a mathematical expression for nine
less than twice a number.
Let x = the number
twice a
number

2x
–9

The expression is 2x – 9.
nine less
than
Section 2.2a
The Product Rule and Power Rules of
Exponents
Definition:
n
a
An exponential expression of the form
n
a
 a  a  a  ...  a where
is
a is any real
n factors of a
number and n is a positive integer. The
base is a and n is the exponent.
Example 1
Identify the base and the exponent in each
expression and evaluate.
a)
34
3 is the base, and 4 is the exponent.
b) (3)4
-3 is the base, and 4 is the exponent.
34  3  3  3  3  81
(3)4  (3)  (3)  (3)  (3)  81
c) 34
3 is the base, and 4 is the exponent.
34  1 34  1 81  81
Product Rule for Exponents
Product Rule: Let a be any real
number and let m and n be positive
integers. Then,
a a  a
m
n
mn
.
Example 2
Find each product.
a) 5  5
2
5
8
4
x

10
x
b)
Solution
a) 52  5  521  53  125
b) 4 x5 10 x8  (4 10)( x58 )  40 x13
Basic Power Rule
Basic Power Rule: Let a be any
real number and let m and n be
positive integers. Then,
a

m n
 a mn .
Example 3
Simplify using the power rule.
a)
4 
 
6 3
b) m
Solution
a)
4
b)
m

6 3

2 5
 463  418
 m25  m10
2 5
Power Rule for a Product
Power Rule for a Product: Let a
and b be any real numbers and let n
be a positive integer. Then,
n n
ab

a
b .
 
n
Example 4
Simplify each expression.
4 5
3
a) (4k )
b)  2w 
Solution
a) (4k )3  43 k 3  64k 3


4 5
b) 2w
 (2)  w
5

4 5
 32w
20
Power Rule for a Quotient
Power Rule for a Quotient: Let a
and b be any real numbers and let n
be a positive integer. Then,
n
a
a
   n , where b  0.
b
b
n
Example 5
Simplify using the power rule for quotients.
4
a)  5 
 
u
b)  v 
3
7
Solution
3
64
4 4
a)  5   53  125
 
3
7
u u



b)  v 
v7
7
Section 2.2b
Combining the Rules of Exponents
Combining the Rules of
Exponents
When we combine the rules of
exponents, we follow the order of
operations.
Example 1
Simplify.
a) (4 f ) (2 f )
2
b) 3  4a b
Solution
2
3
2 2
(4
f
)
(2
f
)

4
f
a)

 16 f  8 f   128 f
3  4a b   3(4)  a   b 
3 3
2
 f 
2
b)
2 3 2

2 3 2
3
3
2
2 2
5
3 2
 3 16a 4b6  48a 4b6
Section 2.3a
Integer Exponents with Real Number
Exponents
Definition:
Zero as an Exponent: If a  0, then
a  1.
0
Example 1
Evaluate.
a) 30
b)
6 0
Solution
a) 30  1
b) 6  1 6  11  1
0
0
Definition:
Negative Exponent: If n is any integer
and a
 0, then a
n
n
1
1
   n .
a
a
n
a
To rewrite an expression of the form
with a positive exponent, take the
reciprocal of the base and make the
exponent positive.
Example 2
Evaluate.
a) (9)
4
b)  
5
2
Solution
2
1
 1
a) ( 9)     
 9  81
2
3
3
4
5
125




b)      
64
5
4
3
Section 2.3b
Integer Exponents with Variable
Bases
Expressions Containing
Variable Bases
The rules that apply to real number
bases also apply when the bases are
variables.
Example 1
Evaluate. Assume the variable does not equal zero.
a)
k
0
b)
Solution
a)
b)
k 1
0
3q  3 1  3
0
3q 0
Example 2
Rewrite the expression with positive exponents.
Assume the variable does not equal zero.
a)
y
3
b)  d 
 
5
Solution
5
a)
1
1
y    5
y
 y
5
4
4
d4 d4
3
d 
b)       4 
3
81
d 
3
4
Definition:
If m and n are any integers and a and b
are real numbers not equal to zero, then
a m bn
 m.
n
b
a
To rewrite the original expression with only positive
exponents, the terms with the negative exponents
“switch” their positions in the fraction.
Example 3
Rewrite the expression with positive exponents. Assume
the variables do not equal zero.
u 1
a) 5
v
9s 10
b)
t4
Solution
u 1 v5 v5
 1
a)
5
v
u
u
9s 10
9

b)
t4
s10t 4
The exponent on t is positive, so do not change its
position in the expression.
Section 2.4
The Quotient Rule
Quotient Rule for Exponents
Quotient Rule for Exponents: If m and n are any
am
mn
integers and a  0, then n  a .
a
To apply the quotient rule, the bases must be the
same. Subtract the exponent of the denominator
from the exponent of the numerator.
Example 1
Simplify. Assume the variable does not equal zero.
74
a) 7 2
n12
b) 4
n
x3
c) 10
x
Solution
74
42
2

7

7
 49
a) 2
7
x3
1
c) 10  x310  x 7  7
x
x
n12
12  4
8

n

n
b)
n4
Example 2
Simplify the expression. Assume the
variables do not equal zero. 16 p 6 q 4
6 p 2 q 3
Solution
16 p 6 q 4 8 6( 2) 43
8 6  2 1
 p
q

p q
2 3
6p q
3
3
8 4
8q
 p q 
3
3 p4
Mid-Chapter Summary
Putting the Rules Together
Example 1
2
 21 p q 

8 4  .
 28 p q 
7
3
Simplify
Assume the variables do
not equal zero.
Solution
Begin by taking the reciprocal of the base to eliminate the
negative on the exponent on the outside of the parentheses.
2
 21 p q 
 28 p q 



8 4 
7 3 
28
p
q
21
p
q 



7
3
8
4

  p15 q 7 
3

2
4
2
4

  p8( 7) q 43 
3

16 30 14
16 p 30

p q 
9
9q14
2
Section 2.5
Scientific Notation
Definition:
n
a
A number is in scientific notation if it is
n
a

10
written in the form
where 1  a  10
and n is an integer.
1  a  10 means that a is a number that has
one nonzero digit to the left of the decimal
point. Here are two numbers in scientific
notation:
8.174 105
2.3 108
Example 1
Write without exponents
a)
3.904  104
b)
1.07 102
Solution
a)
Move the decimal point 4 places to the right. Multiplying 3.904
by a positive power of 10 will make the result larger than
3.904.
4
3.904 10  39, 040
b)
Move the decimal point 2 places to the left. Multiplying 1.07 by
a negative power of 10 will make the result smaller than 1.07.
1.07 102  0.0107
Example 2
Write each number in scientific notation.
a)
52, 000, 000
b)
0.00009
Solution
a)
To write 52, 000, 000 in scientific notation, the decimal point
must go between the 5 and the 2. This will move the decimal
point 6 places.
52,000,000  5.2 106
b)
To write 0.00009 in scientific notation, the decimal point must
go after the 9. This will move the decimal point 5 places.
0.00009  9 105