Learning Outcomes
- Multiply 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] |
try it
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]
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]
try it
example
Multiply: [latex]\left(\Large\frac{3}{4}\normalsize{c}^{3}d\right)\left(12c{d}^{2}\right)[/latex]
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.
