To simplify the given expression [tex]\(\frac{(3m^2n)^3}{mn^4}\)[/tex], follow these steps:
1. Expand the numerator [tex]\((3m^2n)^3\)[/tex]:
[tex]\[
(3m^2n)^3 = 3^3 \cdot (m^2)^3 \cdot n^3
\][/tex]
Compute the powers and the multiplication:
[tex]\[
3^3 = 27,\ (m^2)^3 = m^{2 \cdot 3} = m^6,\ n^3
\][/tex]
Therefore,
[tex]\[
(3m^2n)^3 = 27m^6n^3
\][/tex]
2. Write the initial expression with the expanded numerator:
[tex]\[
\frac{27m^6n^3}{mn^4}
\][/tex]
3. Simplify the denominator [tex]\(mn^4\)[/tex]:
The denominator already is in its simplest form, [tex]\(mn^4\)[/tex].
4. Combine the numerator and the denominator:
[tex]\[
\frac{27m^6n^3}{mn^4}
\][/tex]
5. Simplify the expression by canceling out common factors in the numerator and the denominator:
- For [tex]\(m\)[/tex] terms: Subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
m^6 / m = m^{6 - 1} = m^5
\][/tex]
- For [tex]\(n\)[/tex] terms: Subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
n^3 / n^4 = n^{3 - 4} = n^{-1} = \frac{1}{n}
\][/tex]
Combining these results:
[tex]\[
\frac{27m^6n^3}{mn^4} = 27 \cdot m^5 \cdot \frac{1}{n} = \frac{27m^5}{n}
\][/tex]
So, the given expression [tex]\(\frac{(3m^2n)^3}{mn^4}\)[/tex] simplifies to [tex]\(\frac{27m^5}{n}\)[/tex].
Hence, the correct answer is:
C. [tex]\(\frac{27m^5}{n}\)[/tex]
