Question: Question: If a given quantity of light had a frequency of $5 \times 10^{14}$ s$^{-1}$, what is the light's wavelength?
Question: Convert your answer to nm.
1. **State the problem:** We are given the frequency $f = 5 \times 10^{14}$ s$^{-1}$ of light and need to find its wavelength $\lambda$.
2. **Formula used:** The relationship between the speed of light $c$, frequency $f$, and wavelength $\lambda$ is:
$$ c = \lambda \times f $$
where $c = 3 \times 10^{8}$ m/s (speed of light).
3. **Rearrange the formula to solve for wavelength:**
$$ \lambda = \frac{c}{f} $$
4. **Substitute the known values:**
$$ \lambda = \frac{3 \times 10^{8}}{5 \times 10^{14}} $$
5. **Simplify the fraction:**
$$ \lambda = \frac{\cancel{3} \times 10^{8}}{\cancel{5} \times 10^{14}} \times \frac{1}{\frac{5}{3}} = \frac{3}{5} \times 10^{8 - 14} = 0.6 \times 10^{-6} \text{ meters} $$
6. **Convert meters to nanometers:**
Since $1$ meter $= 10^{9}$ nanometers,
$$ \lambda = 0.6 \times 10^{-6} \times 10^{9} = 0.6 \times 10^{3} = 600 \text{ nm} $$
**Final answer:** The wavelength of the light is $600$ nm.