3.2 Products and Quotients of Monomials

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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