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Multiplying and Dividing Using Scientific Notation

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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.\(300,000\)

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