chapter 2 variables and exponents. section 2.1 simplifying expressions
TRANSCRIPT
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
a) Are like terms?
Yes. Each variable contains the variable p with and exponent of 2. They are -terms.
b) Are like terms?
No. Although each contains the variable f, the exponents are not the same.
2 2 212 , 0.1 , and
5p p p
2p
3 4 0.418 , , and ff f
Combine like terms
We combine like terms using the distributive property. We can add and subtract only those terms that are like terms.
Example 2Combine like terms.
a)
We combine the coefficients of the like terms.
b) 3 3 3 3
3
9 4 3 9 4 3
8 7
w w w w w w w w
w w
3 8 5 1 (3 5) (8 1)
2 7
d d d
d
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(3 2) ( 4 9) 5(3 2) 1( 4 9)
15 10 4 9
15 4 10 9
19 1
n n n n
n n
n n
n
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
2x – 9
The expression is 2x – 9.
twice a
number
nine less
than
Section 2.2a
The Product Rule and Power Rules of Exponents
Definition:
An exponential expression of the form is where is any real
number and n is a positive integer. The base is and n is the exponent.
na
factors of
...n
n a
a a a a a a
a
Example 1
Identify the base and the exponent in each expression and evaluate.
a) 3 is the base, and 4 is the exponent.
b) -3 is the base, and 4 is the exponent.
c) 3 is the base, and 4 is the exponent.
4343 3 3 3 3 81
4( 3)4( 3) ( 3) ( 3) ( 3) ( 3) 81
4 43 1 3 1 81 81
43
Product Rule for Exponents
Product Rule: Let be any real number and let m and n be positive integers. Then,
a
.m n m na a a
Example 2
Find each product.a) b)
Solutiona)
b)
25 55 84 10x x
2 2 1 35 5 5 5 125
5 8 5 8 134 10 (4 10)( ) 40x x x x
Basic Power Rule
Basic Power Rule: Let be any real number and let m and n be positive integers. Then,
a
.nm mna a
Example 3Simplify using the power rule.
a) b)
Solution
a)
b)
364 52m
36 6 3 184 4 4
52 2 5 10m m m
Power Rule for a Product
Power Rule for a Product: Let and be any real numbers and let n be a positive integer. Then,
a
.n n nab a b
b
Example 4
Simplify each expression.a) b)
Solutiona)
b)
3(4 )k 542w
3 3 3 3(4 ) 4 64k k k
5 54 5 4 202 ( 2) 32w w w
Power Rule for a Quotient
Power Rule for a Quotient: Let and be any real numbers and let n be a positive integer. Then,
a
, where 0.n n
n
a ab
b b
b
Example 5
Simplify using the power rule for quotients.
a) b)
Solution
a) b)
34
5
7u
v
3 3
3
4 4 64
5 5 125
7 7
7
u u
v v
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) b)
Solutiona)
b)
2 3(4 ) (2 )f f 22 33 4a b
2 3 2 2 3 3
2 3 5
(4 ) (2 ) 4 2
16 8 128
f f f f
f f f
2 2 22 3 2 2 3
4 6 4 6
3 4 3(4)
3 16 48
a b a b
a b a b
Section 2.3a
Integer Exponents with Real Number Exponents
Definition:
Zero as an Exponent: If then0 1.a 0,a
Example 1
Evaluate.a) b)
Solutiona)
b)
03 06
03 1
0 06 1 6 1 1 1
Definition:
Negative Exponent: If n is any integer
and then
To rewrite an expression of the form with a positive exponent, take the reciprocal of the base and make the exponent positive.
1 1.
nn
na
a a
0,a
na
Example 2
Evaluate.a) b)
Solution
a)
b)
2( 9)3
4
5
22 1 1
( 9)9 81
3 34 5 125
5 4 64
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 1Evaluate. Assume the variable does not equal zero.
a) b)
Solution
a)
b)
0k 03q
0 1k
03 3 1 3q
Example 2Rewrite the expression with positive exponents. Assume the variable does not equal zero.
a) b)
Solution
a)
b)
5y 4
3
d
5
55
1 1y
y y
4 4 4 4
4
3
3 3 81
d d d
d
Definition:
If m and n are any integers and and are real numbers not equal to zero, then
To rewrite the original expression with only positive exponents, the terms with the negative exponents “switch” their positions in the fraction.
.m n
n m
a b
b a
a b
Example 3Rewrite the expression with positive exponents. Assume the variables do not equal zero.
a) b)
Solution
a)
b)
The exponent on t is positive, so do not change its position in the expression.
1
5
u
v
10
4
9s
t
1 5 5
5 1
u v v
v u u
10
4 10 4
9 9s
t s t
Section 2.4
The Quotient Rule
Quotient Rule for Exponents
Quotient Rule for Exponents: If m and n are any
integers and then
To apply the quotient rule, the bases must be the same. Subtract the exponent of the denominator from the exponent of the numerator.
.m
m nn
aa
a0,a
Example 1Simplify. Assume the variable does not equal zero.
a) b) c)
Solution
a) b)
c)
4
2
7
7
12
4
n
n
44 2 2
2
77 7 49
7
1212 4 8
4
nn n
n
3
10
x
x
33 10 7
10 7
1xx x
x x
Example 2
Simplify the expression. Assume the variables do not equal zero.
Solution
6 4
2 3
16
6
p q
p q
6 46 ( 2) 4 3 6 2 1
2 3
44
16 8 8
6 3 3
8 8
3 3
p qp q p q
p q
qp q
p
Mid-Chapter Summary
Putting the Rules Together
Example 1
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.
27 3
8 4
21
28
p q
p q
2 2 27 3 8 48 ( 7) 4 3
8 4 7 3
2 3015 7 30 14
14
21 28 4
28 21 3
4 16 16
3 9 9
p q p qp q
p q p q
pp q p q
q
Section 2.5
Scientific Notation
Definition:
A number is in scientific notation if it is written in the form where and n is an integer.
means that is a number that has one nonzero digit to the left of the decimal point. Here are two numbers in scientific notation:
na10na 1 10a
1 10a a
5 88.174 10 2.3 10
Example 1Write without exponents
a) b)
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.
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.
43.904 10 21.07 10
43.904 10 39,040
21.07 10 0.0107
Example 2Write each number in scientific notation.
a) b)
Solution
a) To write in scientific notation, the decimal point must go between the 5 and the 2. This will move the decimal point 6 places.
b) To write in scientific notation, the decimal point must go after the 9. This will move the decimal point 5 places.
52,000,000 0.00009
652,000,000 5.2 10
50.00009 9 10
52,000,000
0.00009