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MULTIPLYING AND DIVIDING
MONOMIALS
MULTIPLYING MONOMIALS
A MONOMIAL is a number, a variable, or a product
of a number and one or more variables.
EXAMPLES
Numbers: 5, -3, 0.2, 1/3, 1.23, π  CONSTANTS
Variables: x, y, z, a, b, c, k
Product: 5x, -3y, 0.2z, 1/3abc, 125a2x3y
A CONSTANT is a monomial that is a real number.
(no variables)
Which monomials listed above are constants?
MULTIPLYING MONOMIALS
A MONOMIAL is a number a variable, or a product
of a number and one or more variables.
EXAMPLES THAT ARE NOT MONOMIALS
x+y
x
y
Expression involves
addition, not
multiplication.
Expression involves
division, not multiplication
MULTIPLYING MONOMIALS
Remember that the expression 35 is a POWER and it
represents the product you obtain when 3 is used
as a factor 5 times.
So, 35 = 3 • 3 • 3 • 3 • 3 = 243
2 3 • 24 = 2 • 2 • 2
•
2 • 2 • 2 • 2 = 27
MULTIPLYING MONOMIALS
Remember that in the expression 35 is a POWER
and represents the product you obtain when 3 is
used as a factor 5 times.
So, 35 = 3 • 3 • 3 • 3 • 3 = 243
2 3 • 24 = 2 • 2 • 2
•
2 • 2 • 2 • 2 = 27
x2 • x6 = x • x • x • x • x • x • x • x = x8
2
+
6
= 8
MULTIPLYING MONOMIALS
PRODUCT OF POWERS
For any number x and all integers m and n,
xm • xn = xm+n
POWER OF A POWER
For any number x and all integers m and n,
(xm)n = xmn
EXAMPLE: (y2)5 = y10
MULTIPLYING MONOMIALS
PRODUCT OF POWERS
For any number x and all integers m and n,
xm • xn = xm+n
POWER OF A POWER
For any number x and all integers m and n,
(xm)n = xmn
POWER OF A PRODUCT
For any number x and y and any integer m,
(xy)m = xmym
MULTIPLYING MONOMIALS
2x3 • 5x2 = 2 • 5 • x3 • x2 = 10x5
(-6m5)(3m3) = -6 • 3 • m5 • m3 = -18m8
(5y4)2 = 52 • (y4)2 = 25y8
(5a4b2)(4ab7) = 5 • 4 • a4 • a1 • b2 • b7 = 20a5b9
DIVIDING MONOMIALS
x7
X•X•X•X•X•X•X
=
3
x
X•X•X
=
x4
7–3=4
QUOTIENT OF POWERS
For any number x and all integers m and n,
xm
m-n
=
x
n
x
12x8
5
EXAMPLE:
=
4x
3x3
DIVIDING MONOMIALS
POWER OF A QUOTIENT
For any number x and y and any integer m,
x m
xm
= ym
y
a

 b
3
EXAMPLE:
4
a

  4
b

12
DIVIDING MONOMIALS
POWER OF A QUOTIENT
For any number x and y and any integer m,
x m
xm
= ym
y
ZERO EXPONENT
For any NONZERO number x, x0 = 1
NOTE: 00 is undefined.
50 = 1
1870 = 1
(-3)0 = 1
IT’S ALWAYS 1 (except 00)
DIVIDING MONOMIALS
POWER OF A QUOTIENT
For any number x and y and any integer m,
x m
xm
= ym
y
ZERO EXPONENT
For any NONZERO number x, x0 = 1
NEGATIVE EXPONENT
For any NONZERO number x and integer n,
x-n = 1
xn
DIVIDING MONOMIALS
1
5 
5
1
1
3
 2
  
2
 3
NEGATIVE EXPONENT
2y
2x y  2
x
2
ONE MORE TIME!!!
x8 • x5 = x13
(PRODUCT OF POWERS)
(x8)5 = x40
(POWER OF A POWER)
(5x3)2 = 25x6 (POWER OF A PRODUCT)
a7
5

a
a2
m

 n
3
2

m6
  2
n

x0 = 1
5x
3
(QUOTIENT OF POWERS)
5
 3
x
(POWER OF A QUOTIENT)
(ZERO EXPONENT)
(NEGATIVE EXPONENT)
FLASH CARDS
PART 1: IS IT A MONOMIAL?
X
YES
FLASH CARDS
PART 1: IS IT A MONOMIAL?
2
-5x
YES
FLASH CARDS
PART 1: IS IT A MONOMIAL?
3x - 7
NO
FLASH CARDS
PART 1: IS IT A MONOMIAL?
10x
5y
NO
FLASH CARDS
PART 1: IS IT A MONOMIAL?
1
2
YES
FLASH CARDS
PART 2: SIMPLIFYING
2
3
x (x )
5
x
FLASH CARDS
PART 2: SIMPLIFYING
6
2
5m (m )
8
5m
FLASH CARDS
PART 2: SIMPLIFYING
3
2
(-2y )
6
4y
FLASH CARDS
PART 2: SIMPLIFYING
0
2
(3x )
9
FLASH CARDS
PART 2: SIMPLIFYING
2
0
(3x )
1
FLASH CARDS
PART 2: SIMPLIFYING
6
6a
2a2
3a4
FLASH CARDS
PART 2: SIMPLIFYING
5b6
10b0
6
b
2
END