Ultimate Guide to

Multiplying & Dividing

Monomials with

Exponents

Monomials

Multiplying & Dividing Monomials

Applying Exponent Rules to Monomials

Vocabulary

Monomials - a number, a variable, or a product of a number and one or more variables

4x, 20x2yw3, -3, a2b3, and 3yz are all monomials. Constant – a monomial that is a number without a variable. Base – In an expression of the form xn, the base is x. Exponent – In an expression of the form xn, the exponent is

n.

Writing Expressions Using Exponents

Write the expression with exponents

(as multiplication):

8a3b38 ● a ● a ● a ● b ● b ● b =Could the above expression be

written as a power of a product? ( )x

x x x x y y y y

xy xy xy xy xy 4or

Simplify the following expression: (5a2)(a5)

Step 1: Write out the expressions in expanded form.

Step 2: Rewrite using exponents.

Product Rule

5a2 a5 5 a a a a a a a

How many terms are there?

What operation is being performed? Multiplication!

5a2 a5 5 a

7 5a

7

Multiplying Monomials: The Product Rule

4) 3k5mn

4 7k3m

3n

3

5) 12 x2y

3 2xy2 24x

3y

5

21k8m

4n

7

If the monomials have coefficients, multiply those, but still add powers of common bases.

If the monomial inside the parentheses has more than one variable, raise each variable to the outside power using the power of a power rule.

(ab)m = am•bm

(9xy)2 = (-5x)2 = -(5x)2 =

Simplify the following: ( x3 ) 4

Note: 3 x 4 = 12

The monomial is the term inside the parentheses.

1. Multiply the exponents, write the simplified monomial

x3

4

x12

For any number, a, and all integers m and n,

am n

amn .

1) b9

10

b90

2) c3

3

c9

1) 2b9

3

8b27

2) 5c3

3

125c9

3) 7w12

2

49w24

If the monomial inside the parentheses has a coefficient, raise the coefficient to the power, but still

multiply the variable powers.

Dividing Monomials

For all integers “m” and “n” and any nonzero number “a” ……

Let's review the rules.

m

n

a

a

m na When the problems look like this, and the bases are the same, you will subtract the exponents.

0 1a ANY number raised to the zero power is equal to ONE.

na 1na

If the exponent is negative, it is written on the wrong side of the fraction bar, move it to the other side, and change the sign.

1. 3 2 2f g h

fgh

3 1 2 1 2 1f g h 2 1 1f g h

2. 3 5

7

24

6

x y

xy

Subtract the exponents

42x

2y

Reminder: Never finish a problem with negative

exponents

3. 0 4 2

2 3 2

5 t wu

t w u

1

4. 4 5

2 6

27

9

x y

x y

Subtract the exponents

3 2xy

U’s cancelEach other

2t2w

5. 9 3

6 2

x y

xw u

Remember, if the exponent is negative, move it to the other side of the fraction bar and make it positive.

110x 6 2 3w u y

6. 6

8

x

x

6x

8x

NowSubtract

TheExponents

2x

12x

7. 6

3

40

10

x

xFix thenegativeexponent

640x

10

3x

Now divide the coefficients but ADD the exponents

4 9x1

94x

8. 0 8 4 6

6 2

5 x w u

xw u

ANY number raised to the zero power is equal to ONE.

1 7x2w

4u

9. 10 2 16

5 6 4

30

5

x y z

x y z

Fix thenegative

exponents 30

5

5x10x

2 16y z6y

4z

Now divide the coefficients and combine the exponents

65x

4y

20z

10. 54

3

b

c

20

15

b

c

11.

29 3

6

v

w

6v

4w

2 6 5

8 3

( )( )

( )

x y x y

x y12.

7x 7y

Then thedenominator

24x 3y

NowSubtract

TheExponents

4y

17x

First – Simplify the numerator!!

2 8 4

9 2

( )( )

( )

x y x y

x y13.

6x 9y

Then thedenominator

18x 2y

NowSubtract

TheExponents

7y

12x

First – Simplify the numerator!!

14.

45 4 0

4 3 3

7

5

a b c

a b c

Exponents OUTSIDEAnd INSIDE …… Distribute!!

4( 7) 20a 16b 0c

45 16a 12b 12c

Fix yourNegative exponents

4( 7)

4520a

16a16b

12b 12c0c

4547

4a 4b

12cNow

SubtractThe

Exponents

15.

64 3 0

2 2 4

4

3

a b c

a b c

Exponents OUTSIDEAnd INSIDE …… Distribute!!

6( 4) 24a 18b 0c

63 12a 12b 24c

Fix yourNegative exponents

6( 4)

6324a

12a18b

12b 24c0c

6364

12a 6b

24cNow

SubtractThe

Exponents

16. 3 5 4 2

5 1 5 4

(4 )

(4 )

x y

x y

64 10x 8y

204 4x20y

64

20410x 4x

8y

20y

14414x

12y

17. 3 7 6 3

4 1 7 5

(2 )

(2 )

x y

x y

92 21x 18y

202 5x35y

92

20221x 5x

18y

35y

11226x

17y