Let's break down the given expression step by step:
We start with the expression:
[tex]\[
\frac{\left(3 m^{-1} n^2\right)^4}{\left(2 m^{-2} n\right)^3}
\][/tex]
### Step 1: Simplify the Numerator
First, we simplify [tex]\(\left(3 m^{-1} n^2\right)^4\)[/tex]:
[tex]\[
(3 m^{-1} n^2)^4 = (3)^4 \cdot (m^{-1})^4 \cdot (n^2)^4
\][/tex]
Calculating each part:
[tex]\[
(3)^4 = 81
\][/tex]
[tex]\[
(m^{-1})^4 = m^{-4}
\][/tex]
[tex]\[
(n^2)^4 = n^8
\][/tex]
So,
[tex]\[
(3 m^{-1} n^2)^4 = 81 m^{-4} n^8
\][/tex]
### Step 2: Simplify the Denominator
Next, we simplify [tex]\(\left(2 m^{-2} n\right)^3\)[/tex]:
[tex]\[
(2 m^{-2} n)^3 = (2)^3 \cdot (m^{-2})^3 \cdot (n)^3
\][/tex]
Calculating each part:
[tex]\[
(2)^3 = 8
\][/tex]
[tex]\[
(m^{-2})^3 = m^{-6}
\][/tex]
[tex]\[
(n)^3 = n^3
\][/tex]
So,
[tex]\[
(2 m^{-2} n)^3 = 8 m^{-6} n^3
\][/tex]
### Step 3: Create the Fraction
Now we substitute the simplified forms from steps 1 and 2 back into the original expression:
[tex]\[
\frac{81 m^{-4} n^8}{8 m^{-6} n^3}
\][/tex]
### Step 4: Simplify the Fraction Further
Combine the coefficients and exponents:
[tex]\[
\frac{81}{8} \cdot \frac{m^{-4}}{m^{-6}} \cdot \frac{n^8}{n^3}
\][/tex]
We simplify the exponents using the properties of exponents:
[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]
So,
[tex]\[
\frac{81 m^{-4} n^8}{8 m^{-6} n^3} = \frac{81}{8} \cdot m^2 \cdot n^5
\][/tex]
Combining these results, we have:
[tex]\[
\frac{81 m^2 n^5}{8}
\][/tex]
Thus, the final equivalent expression is:
[tex]\[
\boxed{\frac{81 m^2 n^5}{8}}
\][/tex]
