Mathematics » Polynomials II » Use Multiplication Properties of Exponents

# 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:

1. $${\left({y}^{3}\right)}^{6}{\left({y}^{5}\right)}^{4}$$
2. $${\left(-6{x}^{4}{y}^{5}\right)}^{2}.$$

### Solution

1.

$$\begin{array}{cccc}& & & \phantom{\rule{10em}{0ex}}{\left({y}^{3}\right)}^{6}{\left({y}^{5}\right)}^{4}\hfill \\ \text{Use the Power Property.}\hfill & & & \phantom{\rule{10em}{0ex}}{y}^{15}·{y}^{20}\hfill \\ \text{Add the exponents.}\hfill & & & \phantom{\rule{10em}{0ex}}{y}^{35}\hfill \end{array}$$

2.

$$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(-6{x}^{4}{y}^{5}\right)}^{2}\hfill \\ \text{Use the Product to a Power Property.}\hfill & & & \phantom{\rule{4em}{0ex}}{\left(-6\right)}^{2}{\left({x}^{4}\right)}^{2}{\left({y}^{5}\right)}^{2}\hfill \\ \text{Use the Power Property.}\hfill & & & \phantom{\rule{4em}{0ex}}{\left(-6\right)}^{2}\left({x}^{8}\right)\left({y}^{10}\right)\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}36{x}^{8}{y}^{10}\hfill \end{array}$$

## Example

Simplify:

1. $${\left(5m\right)}^{2}\left(3{m}^{3}\right)$$
2. $${\left(3{x}^{2}y\right)}^{4}{\left(2x{y}^{2}\right)}^{3}.$$

### Solution

1.

$$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(5m\right)}^{2}\left(3{m}^{3}\right)\hfill \\ \text{Raise}\phantom{\rule{0.2em}{0ex}}5m\phantom{\rule{0.2em}{0ex}}\text{to the second power.}\hfill & & & \phantom{\rule{4em}{0ex}}{5}^{2}{m}^{2}·3{m}^{3}\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}25{m}^{2}·3{m}^{3}\hfill \\ \text{Use the Commutative Property.}\hfill & & & \phantom{\rule{4em}{0ex}}25·3·{m}^{2}·{m}^{3}\hfill \\ \text{Multiply the constants and add the exponents.}\hfill & & & \phantom{\rule{4em}{0ex}}75{m}^{5}\hfill \end{array}$$

2.

$$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(3{x}^{2}y\right)}^{4}{\left(2x{y}^{2}\right)}^{3}\hfill \\ \text{Use the Product to a Power Property.}\hfill & & & \phantom{\rule{4em}{0ex}}\left({3}^{4}{x}^{8}{y}^{4}\right)\left({2}^{3}{x}^{3}{y}^{6}\right)\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}\left(81{x}^{8}{y}^{4}\right)\left(8{x}^{3}{y}^{6}\right)\hfill \\ \text{Use the Commutative Property.}\hfill & & & \phantom{\rule{4em}{0ex}}81·8·{x}^{8}·{x}^{3}·{y}^{4}·{y}^{6}\hfill \\ \text{Multiply the constants and add the exponents.}\hfill & & & \phantom{\rule{4em}{0ex}}648{x}^{11}{y}^{10}\hfill \end{array}$$

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