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Lesson 1
MULTIPLYING MONOMIALS
What are we going to
do…

Multiply monomials.

Simplify expressions involving
powers of monomials.
Monomial

A monomial is a number, a
variable, or a product of a number
and one or more variables.

An expression involving the division
of variables is not a monomial.

Monomials that are real numbers
are called constants.
Examples of Monomials
1. -5
2. x
3. abc3
4. 5xy2
7
Not a monomial: 4cd3
9ab
Why?
Rule #1:
Product of Powers

To multiply two powers that have
the same base, add the exponents.

For any number x, xm(xn) = xm+n.

x12 ● x5 = x17
Example 1

(5x6)(x3)
= 5(x6x3)
= 5x9
Example 2

(4ab4)(-5a2b3)
= (4)(-5)(aa2)(b4b3)
= -20a3b7
Example 3…on your own!

(2ab5)(-a2b)
= (4)(-1)(aa2)(b5b)
= -4a3b6
Rule #2:
Power of a Power
 To
find the power of a power,
multiply the exponents.
 For
any number a, (am)n = amn.
 (a4)3
= a12
Example 4

[(a2)3]2
= [a6]2
= a12
OR….
= [a] 2*3*2
=a12
Example 5

(x2)4
= x2*4
= a8
Example 6…on your own!

[(32)3]2
= [36]2
= 312
= 531,441
Rule #3:
Power of a Product

To find the power of a product, find the
power of each factor and multiply.

For all numbers x and y,
(xy)m = xmym.

(-2x2y3)3 = (-2)3(x2)3(y3)3 = -8x6y9

Think of it like distributing the
exponent!
Example 7

(3ab)3
= (3)3(a)3(b)3
= 27a3b3
Example 8

(5x2yz3)2
= (5)2(x2)2(y)2(z3)2
= 25x4y2z6
Example 9…on your own!

(-2x3yz4)2
= (-2)2(x3)2(y)2(z4)2
= 4x6y2z8
Simplifying Monomial
Expressions

To simplify an expression involving
monomials, write an equivalent
expression in which:
1. each base appears exactly once
2. there are no powers of powers
3. all fractions are in simplest form
Example 10

[(8g3h4)2]2(2gh5)4
= [(8)2(g3)2(h4)2]2(2)4(g)4(h5)4
= [64g6h8]2(16g4h20)
= (64)2(g6)2(h8)2(16g4h20)
= 4096g12h16(16g4h20)
= (4096)(16)(g12g4)(h16h20)
= 65,536g16h36
Example 11
(ab4)(ab2)
= (aa)(b4b2)
= a2b6
Example 12…on your own!
(-4c4d4)(4cd)
= (-4)(4)(c4c)(d4d)
= -16c5d5
Example 13
(5a2b3c4)(4a2b4c3)
= (5)(4)(a2a2)(b3b4)(c4c3)
= 20a2b7c7
Example 14
(7pq7)2
= (7)2(p)2(q7)2
= 49p2q14
Example 15…on your own!
(5x3)2
= (5)2(x3)2
= 25x6
Example 16
(4cd)2 (-2d2)3
= (4)2(c)2(d)2 (-2)3(d2)3
= 16c2d2
(-8d6)
= (16)(-8)(c2)(d2d6)
= -128c2d8
Example 17
(2ag2)4 (3a2g3)2
= (2)4(a)4(g2)4 (3)2(a2)2(g3)2
= 16a4g8(9a4g6)
= (16)(9)(a4a4)(g8g6)
= 144a8g14
Review
1.
When multiplying powers with the same base,
we ______ the exponents.
add
2.
When raising a power to a power, we
_____________ the exponents.
multiply