If $\log _{3 n} 675 \sqrt{3}=\log _{n} 75$, determine the value of $n^{5}$.
Using exponent laws,

$$
\begin{aligned}
\log _{3 n} 675 \sqrt{3} & =\log _{n} 75 \\
(3 n)^{\log _{3 n} 675 \sqrt{3}} & =(3 n)^{\log _{n} 75} \\
675 \sqrt{3} & =3^{\log _{n} 75} \cdot n^{\log _{n} 75} \\
675 \sqrt{3} & =3^{\log _{n} 75} \cdot 75 \\
9 \sqrt{3} & =3^{\log _{n} 75} \\
3^{2} \cdot 3^{1 / 2} & =3^{\log _{n} 75} \\
3^{5 / 2} & =3^{\log _{n} 75} \\
5 / 2 & =\log _{n} 75 \\
n^{5 / 2} & =75 \\
n^{5} & =75^{2}
\end{aligned}
$$

Therefore, $n^{5}=75^{2}=5625$.

ANSWER: 5625