Multiplying Monomials

Learning Outcomes

Use the power and product properties of exponents to multiply monomials
Use the power and product properties of exponents to simplify monomials

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

Properties of Exponents

If [latex]a,b[/latex] are real numbers and [latex]m,n[/latex] are whole numbers, then

[latex]\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}[/latex]

example

Simplify: [latex]{\left({x}^{2}\right)}^{6}{\left({x}^{5}\right)}^{4}[/latex]

Solution

	[latex]{\left({x}^{2}\right)}^{6}{\left({x}^{5}\right)}^{4}[/latex]
Use the Power Property.	[latex]{x}^{12}\cdot {x}^{20}[/latex]
Add the exponents.	[latex]{x}^{32}[/latex]

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example

Simplify: [latex]{\left(-7{x}^{3}{y}^{4}\right)}^{2}[/latex]

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example

Simplify: [latex]{\left(6n\right)}^{2}\left(4{n}^{3}\right)[/latex]

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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 the n.

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example

Simplify: [latex]{\left(3{p}^{2}q\right)}^{4}{\left(2p{q}^{2}\right)}^{3}[/latex]

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Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

example

Multiply: [latex]\left(4{x}^{2}\right)\left(-5{x}^{3}\right)[/latex]

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try it

example

Multiply: [latex]\left(\Large\frac{3}{4}\normalsize{c}^{3}d\right)\left(12c{d}^{2}\right)[/latex]

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try it

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

	[latex]{\left(-7{x}^{3}{y}^{4}\right)}^{2}[/latex]
Take each factor to the second power.	[latex]{\left(-7\right)}^{2}{\left({x}^{3}\right)}^{2}{\left({y}^{4}\right)}^{2}[/latex]
Use the Power Property.	[latex]49{x}^{6}{y}^{8}[/latex]

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

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

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

Module 1: Whole Numbers