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Multiplying Monomials
(7-1)
Objective: Multiply monomials.
Simplify expressions involving
monomials.
Monomials
A monomial is a number, a variable, or the
product of a number and one or more
variables with nonegative integer exponents.
It has only one term.
An expression that involves division by a
variable is not a monomial.
The monomial 3x is an example of a linear
expression since the exponent of x is 1.
The monomial 2x2 is a nonlinear expression
since the exponent is a positive number other
than 1.
Example 1
Determine whether each expression is a
monomial. Write yes or no. Explain your
reasoning.
a.
17 – c

b.
8f2g

c.
Yes, this is a product of numbers and variables.
¾

d.
No, this expression has two terms.
Yes, this is a constant.
5/
t

No, there is a variable in the denominator.
Check Your Progress
Choose the best answer for the following.

Which expression is a monomial?
A. x5
B. 3p – 1
C. 9x/y
D. c/d
Powers
Recall that an expression of the form xn is
called a power and represents the result of
multiplying x by itself n times.
x is the base, and n is the exponent.
The word power is also used sometimes to
refer to the exponent.
exponent
34 = 3 ∙ 3 ∙ 3 ∙ 3 = 81
base
Product of Powers
By applying the definition of a power, you can find
the product of powers.
Look for a pattern in the exponents.


22 ∙ 24 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 26
43 ∙ 42 = 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 = 45
To multiply two powers that have the same base,
add their exponents.
For any real number a and any integers m and p,
am ∙ ap = am+p.
b3 ∙ b5 = b3+5 = b8
g4 ∙ g6 = g4+6 = g10
Example 2
Simplify each expression.
a.
(r4)(-12r7)
=
=
b.
(1 ∙ -12)(r4 ∙ r7)
-12r11
(6cd5)(5c5d2)
=
=
(6 ∙ 5)(c1 ∙ c5)(d5 ∙ d2)
30c6d7
Check Your Progress
Choose the best answer for the following.
A.
Simplify (5x2)(4x3).
A.
B.
C.
D.
9x5
20x5
20x6
9x6
(5 ∙ 4)(x2 ∙ x3)
Check Your Progress
Choose the best answer for the following.
B.
Simplify 3xy2(-2x2y3).
A.
B.
C.
D.
6xy5
-6x2y6
1x3y5
-6x3y5
(3 ∙ -2)(x1 ∙ x2)(y2 ∙ y3)
Power of a Power
We can use the Product of Powers Property to find
the power of a power.
In the following examples, look for a pattern in the
exponents.


(32)4 = (32)(32)(32)(32) = 32+2+2+2 = 38
(r4)3 = (r4)(r4)(r4) = r4+4+4 = r12
To find the power of a power, multiply the
exponents.
For any real number a and any integers m and p,
(am)p = a m∙p.
(b3)5 = b3∙5 = b15
(g6)7 = g6∙7 = g42
Example 3
Simplify [(23)3]2.
=
=
=
23∙3∙2
218
262,144
Check Your Progress
Choose the best answer for the following.

Simplify [(42)2]3.
A.
B.
C.
D.
47
48
412
410
42∙2∙3
Power of a Product
We can use the Product of Powers Property and
the Power of a Power Property to find the power of
a product.
In the following examples, look for a pattern in the
exponents.


(tw)3 = (tw)(tw)(tw) = (t1 ∙ t1 ∙ t1)(w1 ∙ w1 ∙ w1) = t3w3
(2yz2)3 = (2yz2)(2yz2)(2yz2) = (2)3(y)3(z2)3 = 8y3z6
To find the power of a product, find the power of
each factor and multiply.
For any real numbers a and b and any integer m,
(ab)m = ambm.
(-2xy3)5 = (-2)5(x)5(y3)5 = -32x5y15
Example 4
Express the volume of a cube with side
length 5xyz as a monomial.




V = s3
V = (5xyz)3
V = 53x3y3z3
V = 125x3y3z3
Check Your Progress
Choose the best answer for the following.

Express the surface area of the cube as a
monomial.
A.
B.
C.
D.
8p3q3
24p2q2
6p2q2
8p2q2
SA = 6s2
SA = 6(2pq)2
SA = 6 ∙ 22p2q2
SA = 6 ∙ 4p2q2
Simplify Expressions
We can combine and use these properties
to simplify expressions involving
monomials.
To simplify a monomial expression, write
an equivalent expression in which:



Each variable base appears exactly once.
There are no powers of powers, and
All fractions are in simplest form.
Example 5
Simplify [(8g3h4)2]2(2gh5)4.
=
=
=
=
(8g3h4)4(2gh5)4
84(g3)4(h4)4 ∙ 24(g)4(h5)4
(4096 ∙ 16)(g12 ∙ g4)(h16 ∙ h20)
65,536g16h36
Check Your Progress
Choose the best answer for the following.

Simplify [(2c2d3)2]3(3c5d2)3.
A.
B.
C.
D.
1728c27d24
6c7d5
24c13d10
5c7d21
= (2c2d3)6 ∙ (3c5d2)3
= 26(c2)6(d3)6 ∙ 33(c5)3(d2)3
= (64 ∙ 27)(c12 ∙ c15)(d18 ∙ d6)