8.4.1: Multiplication of Monomials

Learning Outcomes

Multiply monomials

Key words

Product: the answer when two or more terms are multiplied

Properties of Exponents

We learned about and used properties for multiplying expressions with exponents in chapter 1. Let’s summarize them here, then we’ll show some examples that use more than one of the properties.

Properties of Exponents

If $a,b$ are real numbers and $m,n$ are whole numbers, then

$\begin{array}{cccc}\text{Product Property}\hfill & & & \hfill {a}^{m}\cdot {a}^{n}={a}^{m+n}\hfill \\ \text{Power Property}\hfill & & & \hfill {\left({a}^{m}\right)}^{n}={a}^{m\cdot n}\hfill \\ \text{Product to a Power Property}\hfill & & & \hfill {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \end{array}$

example

Simplify: ${\left({x}^{2}\right)}^{6}{\left({x}^{5}\right)}^{4}$

Solution

	${\left({x}^{2}\right)}^{6}{\left({x}^{5}\right)}^{4}$
Use the Power Property: multiply exponents	${x}^{12}\cdot {x}^{20}$
Use the Product Property: add the exponents.	${x}^{32}$

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example

Simplify: ${\left(-7{x}^{3}{y}^{4}\right)}^{2}$

Solution

	${\left(-7{x}^{3}{y}^{4}\right)}^{2}$
Take each factor to the second power.	${\left(-7\right)}^{2}{\left({x}^{3}\right)}^{2}{\left({y}^{4}\right)}^{2}$
Use the Power Property: multiply the exponents	$49{x}^{6}{y}^{8}$

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example

Simplify: ${\left(6n\right)}^{2}\left(4{n}^{3}\right)$

Solution

	${\left(6n\right)}^{2}\left(4{n}^{3}\right)$
Raise $6n$ to the second power.	${6}^{2}{n}^{2}\cdot 4{n}^{3}$
Simplify.	$36{n}^{2}\cdot 4{n}^{3}$
Use the Commutative Property.	$36\cdot 4\cdot {n}^{2}\cdot {n}^{3}$
Multiply the constants and add the exponents.	$144{n}^{5}$

Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to $n$ .

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example

Simplify: ${\left(3{p}^{2}q\right)}^{4}{\left(2p{q}^{2}\right)}^{3}$

Solution

	${\left(3{p}^{2}q\right)}^{4}{\left(2p{q}^{2}\right)}^{3}$
Use the Power of a Product Property.	${3}^{4}{\left({p}^{2}\right)}^{4}{q}^{4}\cdot {2}^{3}{p}^{3}{\left({q}^{2}\right)}^{3}$
Use the Power Property.	$81{p}^{8}{q}^{4}\cdot 8{p}^{3}{q}^{6}$
Use the Commutative Property.	$81\cdot 8\cdot {p}^{8}\cdot {p}^{3}\cdot {q}^{4}\cdot {q}^{6}$
Multiply the constants and add the exponents for each variable.	$648{p}^{11}{q}^{10}$

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Multiplying Monomials

In math, we build on concepts we have already learned. Since a monomial is an algebraic term, we can use the properties for simplifying expressions with exponents to multiply monomials.

example

Multiply: $\left(4{x}^{2}\right)\left(-5{x}^{3}\right)$

Solution

	$\left(4{x}^{2}\right)\left(-5{x}^{3}\right)$
Use the Commutative Property to rearrange the factors.	$4\cdot \left(-5\right)\cdot {x}^{2}\cdot {x}^{3}$
Multiply.	$-20{x}^{5}$

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example

Find the product of $\left(\frac{3}{4}{c}^{3}d\right)\text{ and }\left(12c{d}^{2}\right)$

Solution

(\frac{3}{4} c^{3} d) (12 c d^{2})

$\left(\frac{3}{4}{c}^{3}d\right)\left(12c{d}^{2}\right)$

Use the Commutative Property to rearrange

the factors.

\frac{3}{4} \cdot 12 \cdot c^{3} \cdot c \cdot d \cdot d^{2}

$\frac{3}{4}\cdot 12\cdot {c}^{3}\cdot c\cdot d\cdot {d}^{2}$

Multiply.

9 c^{4} d^{3}

$9{c}^{4}{d}^{3}$

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For more examples of how to use the power and product rules of exponents to simplify and multiply monomials, watch the following video.

Chapter 8: Polynomials

Learning Outcomes

Key words

Properties of Exponents

Properties of Exponents

example

Solution

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example

Solution

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example

Solution

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example

Solution

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Multiplying Monomials

example

Solution

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example

Solution

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