Simplifying Expressions By Applying Several Properties
Simplifying Expressions by Applying Several Properties
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 \(a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b\) are real numbers, and \(m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\) are whole numbers, then
\(\begin{array}{cccccc}\mathbf{\text{Product Property}}\hfill & & & \hfill {a}^{m}·{a}^{n}& =\hfill & {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & \hfill {\left({a}^{m}\right)}^{n}& =\hfill & {a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & & \hfill {\left(ab\right)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \end{array}\)
All exponent properties hold true for any real numbers \(m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\). Right now, we only use whole number exponents.
Example
Simplify:
- \({\left({y}^{3}\right)}^{6}{\left({y}^{5}\right)}^{4}\)
- \({\left(-6{x}^{4}{y}^{5}\right)}^{2}.\)
Solution
-
\(\begin{array}{cccc}& & & \phantom{\rule{10em}{0ex}}{\left({y}^{3}\right)}^{6}{\left({y}^{5}\right)}^{4}\hfill \\ \text{Use the Power Property.}\hfill & & & \phantom{\rule{10em}{0ex}}{y}^{15}·{y}^{20}\hfill \\ \text{Add the exponents.}\hfill & & & \phantom{\rule{10em}{0ex}}{y}^{35}\hfill \end{array}\)
-
\(\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(-6{x}^{4}{y}^{5}\right)}^{2}\hfill \\ \text{Use the Product to a Power Property.}\hfill & & & \phantom{\rule{4em}{0ex}}{\left(-6\right)}^{2}{\left({x}^{4}\right)}^{2}{\left({y}^{5}\right)}^{2}\hfill \\ \text{Use the Power Property.}\hfill & & & \phantom{\rule{4em}{0ex}}{\left(-6\right)}^{2}\left({x}^{8}\right)\left({y}^{10}\right)\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}36{x}^{8}{y}^{10}\hfill \end{array}\)
Example
Simplify:
- \({\left(5m\right)}^{2}\left(3{m}^{3}\right)\)
- \({\left(3{x}^{2}y\right)}^{4}{\left(2x{y}^{2}\right)}^{3}.\)
Solution
-
\(\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(5m\right)}^{2}\left(3{m}^{3}\right)\hfill \\ \text{Raise}\phantom{\rule{0.2em}{0ex}}5m\phantom{\rule{0.2em}{0ex}}\text{to the second power.}\hfill & & & \phantom{\rule{4em}{0ex}}{5}^{2}{m}^{2}·3{m}^{3}\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}25{m}^{2}·3{m}^{3}\hfill \\ \text{Use the Commutative Property.}\hfill & & & \phantom{\rule{4em}{0ex}}25·3·{m}^{2}·{m}^{3}\hfill \\ \text{Multiply the constants and add the exponents.}\hfill & & & \phantom{\rule{4em}{0ex}}75{m}^{5}\hfill \end{array}\)
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\(\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(3{x}^{2}y\right)}^{4}{\left(2x{y}^{2}\right)}^{3}\hfill \\ \text{Use the Product to a Power Property.}\hfill & & & \phantom{\rule{4em}{0ex}}\left({3}^{4}{x}^{8}{y}^{4}\right)\left({2}^{3}{x}^{3}{y}^{6}\right)\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}\left(81{x}^{8}{y}^{4}\right)\left(8{x}^{3}{y}^{6}\right)\hfill \\ \text{Use the Commutative Property.}\hfill & & & \phantom{\rule{4em}{0ex}}81·8·{x}^{8}·{x}^{3}·{y}^{4}·{y}^{6}\hfill \\ \text{Multiply the constants and add the exponents.}\hfill & & & \phantom{\rule{4em}{0ex}}648{x}^{11}{y}^{10}\hfill \end{array}\)
This lesson is part of:
Polynomials II