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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
