## Multiplying and Dividing Using Scientific Notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

## Example

Multiply. Write answers in decimal form: \(\left(4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\right)\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}\right).\)

### Solution

\(\left(4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\right)\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}\right)\) | |

Use the Commutative Property to rearrange the factors. | \(4·2·{10}^{5}·{10}^{-7}\) |

Multiply 4 by 2 and use the Product Property to multiply \({10}^{5}\) by \({10}^{-7}\). | \(8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}\) |

Change to decimal form by moving the decimal two places left. | \(0.08\) |

## Example

Divide. Write answers in decimal form: \(\frac{9\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}}{3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}}.\)

### Solution

\(\frac{9\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}}{3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}}\) | |

Separate the factors. | \(\frac{9}{3}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{{10}^{3}}{{10}^{-2}}\) |

Divide 9 by 3 and use the Quotient Property to divide \({10}^{3}\) by \({10}^{-2}\). | \(3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\) |

Change to decimal form by moving the decimal five places right. | \(300,000\) |