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Chapter 1: Problem 58

\([(1\) femtometer \() /(100\) nanometer \()]=\ldots \ldots \ldots \ldots \ldots\) (a) \(10^{-6}\) (b) \(10^{-8}\) (c) \(10^{24}\) (d) \(10^{-24}\)

Short Answer

Expert verified
The short answer is: \[(b) \: 10^{-8}\]

Step by step solution

01

Understanding the prefixes

A femtometer is \(10^{-15}\) meters and a nanometer is \(10^{-9}\) meters. We need to convert the given lengths into meters using these prefixes.
02

Converting lengths to meters

Convert 1 femtometer to meters by multiplying by the conversion factor for femtometers: \[1 \text{ femtometer} = 1 \times 10^{-15} \text{ meters}\] Convert 100 nanometers to meters by multiplying by the conversion factor for nanometers: \[100 \text{ nanometers} = 100 \times 10^{-9} \text{ meters}\]
03

Divide the lengths to find the ratio

Divide 1 femtometer by 100 nanometers to find the ratio: \[\frac{1 \times 10^{-15} \text{ meters}}{100 \times 10^{-9} \text{ meters}}\]
04

Simplify the expression

Simplify the expression by dividing the numbers and combining the powers of 10: \[\frac{1 \times 10^{-15}}{100 \times 10^{-9}} = \frac{1}{100} \times \frac{10^{-15}}{10^{-9}} = \frac{1}{100} \times 10^{(-15-(-9))} = \frac{1}{100} \times 10^{-6}\]
05

Convert the fraction to a power of 10

Convert the fraction \(\frac{1}{100}\) to a power of 10 by changing 100 to \(10^2\): \[\frac{1}{10^2} \times 10^{-6} = 10^{-2} \times 10^{-6}\]
06

Combine powers of 10

Combine the powers of 10 to get the final answer: \[10^{-2} \times 10^{-6} = 10^{(-2-6)} = 10^{-8}\]
07

Match the answer to one of the given choices

Our final answer is \(10^{-8}\) which matches choice (b). Therefore, the correct answer is: \[(b) \: 10^{-8}\]

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