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3.2 Products and Quotients of Monomials
BobsMathClass.Com Copyright © 2010 All Rights Reserved.1
3 4 5(5)(2) x x y y
Your Turn Problem #1
3 2 8 2 5Find the product of 6x yz 2x y z11 3 7Answer: 12x y z
4 9 10x yAnswer:
Product Rule of Exponents
To multiply two like variable factors: 3 4 7
43
a a a a a a a a a aaa
Multiplication can always be performed between two factors. Exponents will change when two like variable factors are being multiplied.
Procedure: To Multiply monomialsStep 1. Multiply numerical coefficientsStep 2. Multiply like variables one at a time, in alphabetical order.
3 4 5 Multiply 5xExample 1. y 2xy
Solution:
n m n mIf b is any real number, and m and n are positive integers, then b b b .
Multiplying Monomials
3.2 Products and Quotients of Monomials
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2 5 3 2 32 Multiply Example 2. a b 6a a b
3
Solution: 2 3 2 5 326 a a a b b
3
Answer: 7 84a b
Your Turn Problem #2
3 4 2 5 43Find the product of 12a bc a c b
4
5 5 9Answer: 9a b c
An exponent written immediately following a parenthesis indicates the number of times the term within the parentheses is being multiplied by itself.
32 2 2 2 6Example: x x x x x
Power to a Power Rule of Exponents
mn n mIf b is any real number, and m and n are positive integers, then b b .
Examples: 53 15x x
7 7a aRecall: If no exponent is shown, the understood exponent is 1.
1
1
a a
3 3
3.2 Products and Quotients of Monomials
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xm n mx nxIf a and b are real numbers, and n is a positive integer, then a b a b .
Examples: 34 5 12 15x y x y
43 5 4 12 4 20 12 4 202a bc 2 a b c 16a b c
Combining the two rules for exponents.
Procedure: To simplify expressions with an exponent outside and following parenthesis:
Step 1: Multiply all exponents inside parenthesis by the exponent outside parenthesis.
Step 2: Write the product of Step 1 as the exponent of each variable in the answer.
Step 3: Multiply out the numerical coefficient. 45 2 Simplify 3x y (i.e. raise to the indicaExample 3. ted p ower)
Solution: 4 20 83 x y
Answer: 20 881x y
xm n mx nxUse the rule: a b a b , then simplify.
Your Turn Problem #3
57Simplify: 2a b
35 5Answer: 32a b
3.2 Products and Quotients of Monomials
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Note: When a negative factor is inside the parentheses, and the exponent on the outside is:
even: the result is positiveodd: the result is negative
Examples:
42 8
52 10
x x
x x
32 3 Simplify Exampl 3xe . y4
Solution: 3 6 9( 3) x y
Answer: 6 927x y
xm n mx nxUse the rule: a b a b , then simplify.
Your Turn Problem #4
33Simplify: 5a b
9 3Answer: 125a b
3.2 Products and Quotients of Monomials
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Rewrite both numerator and denominator without exponents.
Divide out pairs of identical factors, one from the numerator and one from the denominator. Each factor of the pair is lined out and converted to an understood “1”.
7
3
x x x x x x x xx x xx
4 (Since there are still 4 x's left in theAnswer: x numer ator)
Dividing Monomials
Recall from arithmetic, a fraction that is equal to 1 contains a numerator that is equal to its denominator. For example:
3
3
6 x y1, 1,
6 x y
Before we give some formal rules for dividing monomials, let’s perform the following with our understanding of exponents.
7
3
xSimplifExample. y
x
Note: When the denominator equals 1, it does not need to be written.
7i.e. 7
1
2
4
a Simplify Example
a.
2
4
a a aa a a aa
2
1Answer:
a
When all factors in the numerator divide out, the numerator equals “1” which must be written.
3.2 Products and Quotients of Monomials
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Quotient Rule of Exponents
If b is any nonzero real number, and m and n are positive integers, then
m, if nb
1
b
b .2 mnn
m
n, if mbb
b .1 nm
n
m
n, if m1b
b .3 n
m
In other words, find the difference between the exponents. Keep the variable where exponent is larger. If the exponent in the numerator is larger, keep the variable in the numerator. If the exponent in the denominator is larger, keep the variable in the denominator.
53
8
xx
x )a Examples: 58
3
x
1
x
x )b 1
x
x )c 3
3
In example c, anything divided by itself equals 1.
3.2 Products and Quotients of Monomials
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Procedure: To divide two monomialsStep 1. Reduce the numerical coefficients.Step 2. Taking each variable type separately, divide out as in the previous
slides.
7
3
32x Simplify Example .
4x5
Step 1. Reduce coefficients8
Step 2. Subtract exponents for like variables
7 38x
Answer: 48x
Your Turn Problem #58
3
56xSimplify:
14x
5Answer: 4x
3.2 Products and Quotients of Monomials
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6 3 5
3 3
144a b cSimplifyExamp
12le
a c6.
b
12
1. Simplify coefficients
2. Subtract the exponents for each variable
3 0 412a b c By definition any real number with an exponent of 0 is equal to 1.
Therefore the answer is: 3 412a c
Your Turn Problem #68 3
2
28a bcSimplify:
14abc
7Answer: 2a c
3.2 Products and Quotients of Monomials
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1. Simplify coefficients.
2. Subtract the exponents for each variable
Example 7. Simplify:yx21
yx148
35
3
2
x3
y2Therefore the answer is:
Your Turn Problem #7
95
42
yx35
yx25Simplify:
Answer: 53yx7
5
The End.B.R. 12-15-07