Question

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

Short answer

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

Step-by-step solution

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.

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: \[100 \text{ nanometers} = 100 \times 10^{-9} \text{ meters}\]

Divide the lengths to find the ratio

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

Simplify the expression

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

Convert the fraction to a power of 10

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

Combine powers of 10

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

Match the answer to one of the given choices

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

Key concepts

SI Units

SI units, also known as the International System of Units, is the standard system of measurement used worldwide. It consists of seven base units from which all other units of measure are derived, ensuring uniformity and consistency across global measurements. For instance:

• The meter is the base unit of length.
• The kilogram is the base unit for mass.
• The second is the base unit of time.

In the context of unit conversion, understanding SI units allows for easy translation of measurement values by using standard prefixes, such as "kilo," "centi," or "nano," which denote powers of ten.
In this exercise, you are dealing with femtometers and nanometers, each of which is associated with its own power of ten in the SI unit system. A femtometer represents a length in terms of meters as \(10^{-15}\) meters, while a nanometer represents \(10^{-9}\) meters. This allows us to convert these small-scale measurements into SI units, providing a clearer understanding for calculations and comparisons.

Powers of Ten

Powers of ten are a crucial concept in mathematics whenever dealing with very large or very small numbers. They provide a convenient way to represent, compare, and calculate such numbers efficiently. In essence, any power of ten can be written as \(10^n\), where \(n\) is the exponent denoting how many times 10 is multiplied by itself.
This is particularly useful when converting between units with SI prefixes. For example:

• A nanometer (\(nm\)) can be expressed as \(10^{-9}\) meters.
• A femtometer (\(fm\)) equates to \(10^{-15}\) meters.

Understanding how to manipulate powers of ten, including multiplying and dividing exponential forms, allows you to simplify expressions easily. In this problem, we used powers of ten to convert and compare lengths in different units and obtained the result by straightforwardly applying the rules of exponents: \(a^m / a^n = a^{m-n}\).
This process helps in deriving a simple numerical expression from seemingly complex unit comparisons.

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or small to be conveniently written in decimal form. It employs powers of ten and is particularly useful for simplifying calculations, especially when dealing with SI units.
A number in scientific notation follows the form \(a \times 10^n\), where \(a\) is a number between 1 and 10, and \(n\) is an integer representing the power of ten.

• For example, 0.000000001 can be written as \(1 \times 10^{-9}\).
• Likewise, 1,000,000 becomes \(1 \times 10^6\).

In the exercise, scientific notation was not only essential for converting units but also simplifying the expression to find the ratio of lengths in the problem context. Through scientific notation, you conveniently manipulate the size of numbers without losing precision, particularly helpful in fields like physics and engineering, where SI units like nanometers and femtometers are common.
By simplifying the fraction \(\frac{1}{100} \times 10^{-6}\), employing basic multiplication rules for exponents, you ultimately presented the solution in its correct scientific format, delivering a concise and practical answer.