## Simplifying Expressions Using the Product to a Power Property

Contents

We will now look at an expression containing a product that is raised to a power. Look for a pattern.

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

What does this mean? | \(2x·2x·2x\) |

We group the like factors together. | \(2·2·2·x·x·x\) |

How many factors of 2 and of \(x?\) | \({2}^{3}·{x}^{3}\) |

Notice that each factor was raised to the power. | \({\left(2x\right)}^{3}\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}{2}^{3}·{x}^{3}\) |

We write: | \({\left(2x\right)}^{3}\) \({2}^{3}·{x}^{3}\) |

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

### Definition: Product to a Power Property of Exponents

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

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

\(\begin{array}{ccc}\hfill {\left(2·3\right)}^{2}& \stackrel{?}{=}& {2}^{2}·{3}^{2}\hfill \\ \hfill {6}^{2}& \stackrel{?}{=}& 4·9\hfill \\ \hfill 36& =& 36✓\hfill \end{array}\)

## Example

Simplify: \({\left(-11x\right)}^{2}.\)

### Solution

\({\left(-11x\right)}^{2}\) | |

Use the Power of a Product Property, \({\left(ab\right)}^{m}={a}^{m}{b}^{m}.\) | |

Simplify. | \(121{x}^{2}\) |

## Example

Simplify: \({\left(3xy\right)}^{3}.\)

### Solution

\({\left(3xy\right)}^{3}\) | |

Raise each factor to the third power. | |

Simplify. | \(27{x}^{3}{y}^{3}\) |

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