To simplify the given expression:
[tex]\[
\frac{(3m^2 n)^3}{m n^4}
\][/tex]
we can follow these steps:
1. Simplify the numerator: Apply the power of a product property [tex]\((ab)^c = a^c b^c\)[/tex]:
[tex]\[
(3m^2 n)^3 = 3^3 (m^2)^3 n^3 = 27m^6 n^3
\][/tex]
So, the expression now looks like:
[tex]\[
\frac{27m^6 n^3}{m n^4}
\][/tex]
2. Simplify the fraction by dividing the numerator by the denominator. This involves subtracting the exponents of like bases:
[tex]\[
\frac{27m^6 n^3}{m n^4} = 27 \frac{m^6}{m^1} \frac{n^3}{n^4} = 27 m^{6-1} n^{3-4} = 27 m^5 n^{-1}
\][/tex]
3. Rewrite with positive exponents: [tex]\(n^{-1} = \frac{1}{n}\)[/tex], so we have:
[tex]\[
27 m^5 n^{-1} = \frac{27m^5}{n}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{27 m^5}{n}
\][/tex]
Therefore, the equivalent expression corresponds to option B.
