Sure, let's solve the given expression step-by-step:
Given expression:
[tex]\[
\frac{\left(3 m^{-1} n^2\right)^4}{\left(2 m^{-2} n\right)^3}
\][/tex]
First, let's simplify the numerator:
[tex]\[
(3 m^{-1} n^2)^4
\][/tex]
We apply the exponent to each term inside the parentheses:
[tex]\[
= 3^4 \cdot (m^{-1})^4 \cdot (n^2)^4
\][/tex]
Calculating each term individually:
[tex]\[
= 81 \cdot m^{-4} \cdot n^8
\][/tex]
So, the numerator becomes:
[tex]\[
81 m^{-4} n^8
\][/tex]
Now, let's simplify the denominator:
[tex]\[
(2 m^{-2} n)^3
\][/tex]
We apply the exponent to each term inside the parentheses:
[tex]\[
= 2^3 \cdot (m^{-2})^3 \cdot n^3
\][/tex]
Calculating each term individually:
[tex]\[
= 8 \cdot m^{-6} \cdot n^3
\][/tex]
So, the denominator becomes:
[tex]\[
8 m^{-6} n^3
\][/tex]
Now, putting the simplified numerator and denominator together, we have:
[tex]\[
\frac{81 m^{-4} n^8}{8 m^{-6} n^3}
\][/tex]
To simplify this fraction, we divide the coefficients:
[tex]\[
= \frac{81}{8}
\][/tex]
Next, we simplify the exponents of [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
[tex]\[
\frac{m^{-4}}{m^{-6}} = m^{-4 - (-6)} = m^{-4 + 6} = m^2
\][/tex]
[tex]\[
\frac{n^8}{n^3} = n^{8-3} = n^5
\][/tex]
Combining all the simplified parts, we get:
[tex]\[
\frac{81}{8} \cdot m^2 \cdot n^5
\][/tex]
Thus, the equivalent expression is:
[tex]\[
\frac{81 m^2 n^5}{8}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{81 m^2 n^5}{8}}
\][/tex]
