Sure, let's simplify the given quotient step-by-step to find which option matches the given fraction:
The given quotient is:
[tex]\[
\frac{\sqrt{16}}{3 \sqrt{2}}
\][/tex]
First, simplify the numerator and the denominator:
1. Simplify the numerator:
[tex]\[
\sqrt{16} = 4
\][/tex]
So, the fraction now looks like:
[tex]\[
\frac{4}{3 \sqrt{2}}
\][/tex]
2. Simplify the denominator:
[tex]\[
3 \sqrt{2}
\][/tex]
So, the fraction remains:
[tex]\[
\frac{4}{3 \sqrt{2}}
\][/tex]
Now, we need to rationalize the denominator, which involves removing the square root from the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator, which in this case is [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
\frac{4}{3 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4 \sqrt{2}}{3 \cdot 2}
\][/tex]
Simplify the denominator:
[tex]\[
3 \cdot 2 = 6
\][/tex]
Thus, the fraction becomes:
[tex]\[
\frac{4 \sqrt{2}}{6}
\][/tex]
Next, simplify the fraction by dividing both the numerator and the denominator by 2:
[tex]\[
\frac{4 \sqrt{2}}{6} = \frac{2 \sqrt{2}}{3}
\][/tex]
Now, compare this simplified form with the given options:
A. [tex]\(\sqrt{8}\)[/tex]
B. [tex]\(2 \sqrt{2}\)[/tex]
C. [tex]\(\frac{2 \sqrt{2}}{3}\)[/tex]
D. [tex]\(\frac{\sqrt{13}}{3}\)[/tex]
We find that the simplified form [tex]\(\frac{2 \sqrt{2}}{3}\)[/tex] matches option C.
Therefore, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
