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Convert to decimal form: $6.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}.$
Convert to decimal form: $1.3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}.$
1,300
Convert to decimal form: $9.25\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}.$
92,500
The steps are summarized below.
To convert scientific notation to decimal form:
Convert to decimal form: $8.9\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}.$
Determine the exponent, n , on the factor 10. | |
Since the exponent is negative, move the decimal point 2 places to the left. | |
Add zeros as needed for placeholders. |
Convert to decimal form: $1.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}.$
0.00012
Convert to decimal form: $7.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}.$
0.075
Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.
Multiply. Write answers in decimal form: $\left(4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\right)\left(2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-7}}\right).$
$\begin{array}{cccc}& & & \left(4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\right)\left(2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-7}}\right)\hfill \\ \\ \\ \text{Use the Commutative Property to rearrange the factors.}\hfill & & & 4\xb72\xb7{10}^{5}\xb7{10}^{\mathrm{-7}}\hfill \\ \\ \\ \text{Multiply.}\hfill & & & 8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}\hfill \\ \\ \\ \text{Change to decimal form by moving the decimal two places left.}\hfill & & & 0.08\hfill \end{array}$
Multiply $\left(3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}\right)\left(2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-8}}\right)$ . Write answers in decimal form.
0.06
Multiply $\left(3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}\right)\left(3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-1}}\right)$ . Write answers in decimal form.
0.009
Divide. Write answers in decimal form: $\frac{9\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}}{3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}}.$
$\begin{array}{cccc}& & & \frac{9\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}}{3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}}\hfill \\ \\ \\ \text{Separate the factors, rewriting as the product of two fractions.}\hfill & & & \frac{9}{3}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\frac{{10}^{3}}{{10}^{\mathrm{-2}}}\hfill \\ \\ \\ \text{Divide.}\hfill & & & 3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\hfill \\ \\ \\ \text{Change to decimal form by moving the decimal five places right.}\hfill & & & \mathrm{300,000}\hfill \end{array}$
Divide $\frac{8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}}{2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-1}}}$ . Write answers in decimal form.
400,000
Divide $\frac{8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}}{4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-2}}}$ . Write answers in decimal form.
20,000
Access these online resources for additional instruction and practice with integer exponents and scientific notation:
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