## Simplifying Expressions Using the Power Property of Exponents Continued

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: | \({\left({x}^{2}\right)}^{3}\) \({x}^{2\cdot 3}\) \({x}^{6}\) |

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

### Definition: Power Property of Exponents

If \(a\) is a real number and \(m,n\) are whole numbers, then

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

An example with numbers helps to verify this property.

\(\begin{array}{ccc}\hfill {\left({5}^{2}\right)}^{3}& \stackrel{?}{=}& {5}^{2·3}\hfill \\ \hfill {\left(25\right)}^{3}& \stackrel{?}{=}& {5}^{6}\hfill \\ \hfill 15,625& =& 15,625✓\hfill \end{array}\)

## Example

Simplify:

- \(\phantom{\rule{0.2em}{0ex}}{\left({x}^{5}\right)}^{7}\)
- \(\phantom{\rule{0.2em}{0ex}}{\left({3}^{6}\right)}^{8}\)

### Solution

\({\left({x}^{5}\right)}^{7}\) | |

Use the Power Property, \({\left({a}^{m}\right)}^{n}={a}^{m·n}.\) | |

Simplify. | \({x}^{35}\) |

\({\left({3}^{6}\right)}^{8}\) | |

Use the Power Property, \({\left({a}^{m}\right)}^{n}={a}^{m·n}.\) | |

Simplify. | \({3}^{48}\) |

### Optional Video: Exponent Properties II

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