Let's simplify the given expression step-by-step.
The given expression is:
[tex]\[
\frac{(3 m^2 n)^3}{m n^4}
\][/tex]
1. First, expand the numerator [tex]\((3 m^2 n)^3\)[/tex]:
- Apply the exponent to each term inside the parentheses:
[tex]\[
(3 m^2 n)^3 = 3^3 \cdot (m^2)^3 \cdot (n)^3
\][/tex]
2. Calculate each individual term:
- [tex]\(3^3 = 27\)[/tex]
- [tex]\((m^2)^3 = m^{2 \cdot 3} = m^6\)[/tex]
- [tex]\(n^3\)[/tex] remains [tex]\(n^3\)[/tex]
Therefore, the numerator becomes:
[tex]\[
27 m^6 n^3
\][/tex]
3. Write out the fraction with the simplified numerator:
[tex]\[
\frac{27 m^6 n^3}{m n^4}
\][/tex]
4. Simplify the fraction:
- Simplify the [tex]\(m\)[/tex] terms: [tex]\(\frac{m^6}{m} = m^{6-1} = m^5\)[/tex]
- Simplify the [tex]\(n\)[/tex] terms: [tex]\(\frac{n^3}{n^4} = n^{3-4} = n^{-1}\)[/tex]
Therefore, the fraction simplifies to:
[tex]\[
27 m^5 n^{-1}
\][/tex]
5. Convert the negative exponent:
- [tex]\(n^{-1} = \frac{1}{n}\)[/tex]
Thus, the simplified expression is:
[tex]\[
\frac{27 m^5}{n}
\][/tex]
The equivalent expression to [tex]\(\frac{(3 m^2 n)^3}{m n^4}\)[/tex] is:
[tex]\[
\boxed{\frac{27 m^5}{n}}
\][/tex]
So, the correct answer is:
A. [tex]\(\frac{27 m^5}{n}\)[/tex]
