Of course! Let's delve into finding the wavelength of a gamma ray with a frequency of [tex]\(1.0 \times 10^{19}\)[/tex] Hz and a given speed of light [tex]\( c = 3.0 \times 10^8 \)[/tex] m/s.
The relationship between the wavelength ([tex]\(\lambda\)[/tex]), the speed of light ([tex]\(c\)[/tex]), and the frequency ([tex]\(f\)[/tex]) is given by the following equation:
[tex]\[
\lambda = \frac{c}{f}
\][/tex]
Plugging in the provided values:
[tex]\[
c = 3.0 \times 10^8 \, \text{m/s}
\][/tex]
[tex]\[
f = 1.0 \times 10^{19} \, \text{Hz}
\][/tex]
We can now substitute these values into the equation:
[tex]\[
\lambda = \frac{3.0 \times 10^8}{1.0 \times 10^{19}}
\][/tex]
Next, we simplify the division:
[tex]\[
\lambda = 3.0 \times 10^8 \, \text{m/s} \div 1.0 \times 10^{19} \, \text{Hz}
\][/tex]
This division of the coefficients gives:
[tex]\[
= 3.0 \div 1.0 = 3.0
\][/tex]
And the division of the powers of ten is:
[tex]\[
10^8 \div 10^{19} = 10^{8-19} = 10^{-11}
\][/tex]
Combining these results, we have:
[tex]\[
\lambda = 3.0 \times 10^{-11} \, \text{m}
\][/tex]
Thus, the wavelength of the gamma ray is:
[tex]\[
3.0 \times 10^{-11} \, \text{m}
\][/tex]
The coefficient and the exponent are:
[tex]\[
(3.0, -11)
\][/tex]
So, the wavelength of the gamma ray is [tex]\(3.0 \times 10^{-11}\)[/tex] meters.
