If n is a natural number, then 10-n = 1 10n . Negative powers of 10 can be used to write

Chapter 4, Problem 71

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If n is a natural number, then \(10^{-n}=\frac{1}{10^{n}}\) Negative powers of 10 can be used to write the decimal part of Hindu-Arabic numerals in expanded form. For example,

\(\begin{aligned} 0.8302 &=\left(8 \times 10^{-1}\right)+\left(3 \times 10^{-2}\right)+\left(0 \times 10^{-3}\right)+\left(2 \times 10^{-4}\right) \\ &=\left(8 \times \frac{1}{10^{1}}\right)+\left(3 \times \frac{1}{10^{2}}\right)+\left(0 \times \frac{1}{10^{3}}\right)+\left(2 \times \frac{1}{10^{4}}\right) \\ &=\left(8 \times \frac{1}{10}\right)+\left(3 \times \frac{1}{100}\right)+\left(0 \times \frac{1}{1000}\right)+\left(2 \times \frac{1}{10,000}\right) \end{aligned} \).

In Exercises 65–72, express each expanded form as a Hindu-Arabic numeral.

\(\begin{aligned}\left(3 \times 10^{4}\right)+\left(7 \times 10^{2}\right)+\left(5 \times 10^{-2}\right) &+\left(8 \times 10^{-3}\right) \\+&\left(9 \times 10^{-5}\right)\end{aligned}\)

Text Transcription:

10^-n=1/10^n

0.8302=(8 times 10^-1)+(3 times 10^-2)+(0 times 10^-3)+(2 times 10^-4)

=(8 times 1/10^1)+(3 times 1/10^2)+(0 times 1/10^3)+(2 times 1/10^4)

=(8 times 1/10)+(3 times 1/100)+(0 times 1/1000)+(2 times 1/10,000)

(3 times 10^4)+(7 times 10^2)+(5 times 10^-2)+(8 times 10^-3)+(9 times 10^-5)