How many meters are in 108 nanometers? Express the answer in correct scientific notation.
$108\text{ nm} = [?] \times 10^{[?]} \text{ m}$
To convert nanometers to meters, we must use the definition of the prefix 'nano'. The prefix 'nano' (n) represents a factor of \(10^{-9}\), meaning that \(1 \text{ nm} = 10^{-9} \text{ m}\). We are asked to convert \(108 \text{ nm}\) into this scientific notation format: \(a \times 10^b \text{ m}\), where the coefficient \(a\) satisfies \(1 \le a < 10\).
We begin with the value \(108 \text{ nm}\). Applying the conversion factor, we have:
\[108 \text{ nm} = 108 \times 10^{-9} \text{ m}\]
To write this in scientific notation, we express the numerical coefficient \(108\) as a product of a number between \(1\) and \(10\) and a power of \(10\). Specifically, \(108 = 1.08 \times 10^2\).
Now, we substitute the expression for \(108\) back into our equation:
\[108 \text{ nm} = (1.08 \times 10^2) \times 10^{-9} \text{ m}\]
Using the laws of exponents, specifically \(10^m \times 10^n = 10^{m+n}\), we combine the powers of \(10\):
\[10^2 \times 10^{-9} = 10^{2 + (-9)} = 10^{-7}\]
Therefore, the conversion is:
\[108 \text{ nm} = 1.08 \times 10^{-7} \text{ m}\]
Comparing this to the desired form \(a \times 10^b \text{ m}\), we identify the coefficient \(a = 1.08\) and the exponent \(b = -7\).
Scientific notation is essential for expressing extremely small or large quantities in fields like physics and chemistry. For example, when measuring the wavelength of visible light, which is often in the range of hundreds of nanometers (like the 108 nm used here), scientists use scientific notation to maintain precision and simplify calculations. Understanding how to convert these units allows researchers to relate microscopic measurements to standard SI units like the meter, which is fundamental for engineering and scientific analysis.