## Simplifying Expressions Using the Power Property For Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

What does this mean? How many factors altogether? | |

So we have | |

Notice that 6 is the product of the exponents, 2 and 3. |

We write:

\(\begin{array}{c}\hfill {\left({x}^{2}\right)}^{3}\hfill \\ \hfill {x}^{2·3}\hfill \\ \hfill {x}^{6}\hfill \end{array}\)

We multiplied the exponents. This leads to the **Power Property for Exponents.**

### Power Property for Exponents

If \(a\) is a real number, and \(m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\) are whole numbers, then

\({\left({a}^{m}\right)}^{n}={a}^{m·n}\)

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

\(\begin{array}{ccc}\hfill {\left({3}^{2}\right)}^{3}& \stackrel{?}{=}\hfill & {3}^{2·3}\hfill \\ \hfill {\left(9\right)}^{3}& \stackrel{?}{=}\hfill & {3}^{6}\hfill \\ \hfill 729& =\hfill & 729\phantom{\rule{0.2em}{0ex}}\text{✓}\hfill \end{array}\)

## Example

Simplify: (a) \({\left({y}^{5}\right)}^{9}\) (b) \({\left({4}^{4}\right)}^{7}.\)

### Solution

Use the power property, (a)^{m} = ^{n}a.^{m·n} | |

Simplify. |

Use the power property. | |

Simplify. |