To simplify the given expression [tex]\(\frac{\left(3 m^2 n\right)^3}{m n^4}\)[/tex], we will follow a step-by-step process.
First, let's break down the numerator [tex]\((3 m^2 n)^3\)[/tex]:
[tex]\[
(3 m^2 n)^3 = 3^3 \cdot (m^2)^3 \cdot n^3
\][/tex]
Now calculate each part separately:
[tex]\[
3^3 = 27
\][/tex]
[tex]\[
(m^2)^3 = m^{2 \times 3} = m^6
\][/tex]
[tex]\[
n^3 = n^3
\][/tex]
So the numerator becomes:
[tex]\[
27 m^6 n^3
\][/tex]
Next, let's look at the denominator [tex]\(m n^4\)[/tex]:
[tex]\[
m n^4
\][/tex]
So the expression now looks like:
[tex]\[
\frac{27 m^6 n^3}{m n^4}
\][/tex]
To simplify, divide the terms with the same base:
[tex]\[
\frac{27 m^6 n^3}{m^1 n^4}
\][/tex]
This can be written as:
[tex]\[
27 \cdot \frac{m^6}{m^1} \cdot \frac{n^3}{n^4}
\][/tex]
Simplify each fraction:
[tex]\[
\frac{m^6}{m^1} = m^{6-1} = m^5
\][/tex]
[tex]\[
\frac{n^3}{n^4} = n^{3-4} = n^{-1} = \frac{1}{n}
\][/tex]
Combine these results together:
[tex]\[
27 \cdot m^5 \cdot \frac{1}{n} = \frac{27 m^5}{n}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{27 m^5}{n}}
\][/tex]
