To solve the expression [tex]\(\frac{\left(3 m^{-1} n^2\right)^4}{\left(2 m^{-2} n\right)^3}\)[/tex], let's break it down step-by-step.
### Step 1: Expand the numerator
The numerator is [tex]\((3 m^{-1} n^2)^4\)[/tex].
We can break it down as follows:
[tex]\[
(3 m^{-1} n^2)^4 = 3^4 \cdot (m^{-1})^4 \cdot (n^2)^4
\][/tex]
Calculate each part individually:
[tex]\[
3^4 = 81
\][/tex]
[tex]\[
(m^{-1})^4 = m^{-4}
\][/tex]
[tex]\[
(n^2)^4 = n^8
\][/tex]
So, the numerator becomes:
[tex]\[
81 m^{-4} n^8
\][/tex]
### Step 2: Expand the denominator
The denominator is [tex]\((2 m^{-2} n)^3\)[/tex].
We can break it down as follows:
[tex]\[
(2 m^{-2} n)^3 = 2^3 \cdot (m^{-2})^3 \cdot n^3
\][/tex]
Calculate each part individually:
[tex]\[
2^3 = 8
\][/tex]
[tex]\[
(m^{-2})^3 = m^{-6}
\][/tex]
[tex]\[
n^3 = n^3
\][/tex]
So, the denominator becomes:
[tex]\[
8 m^{-6} n^3
\][/tex]
### Step 3: Combine and simplify
Now we need to combine the numerator and the denominator:
[tex]\[
\frac{81 m^{-4} n^8}{8 m^{-6} n^3}
\][/tex]
This can be simplified by dividing the coefficients and applying the properties of exponents:
[tex]\[
\frac{81}{8} \cdot \frac{m^{-4}}{m^{-6}} \cdot \frac{n^8}{n^3}
\][/tex]
### Step 4: Simplify the exponents
Using the properties of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex] and [tex]\((a^m)^n = a^{mn}\)[/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]
### Step 5: Write the final form
Combining these results, we get:
[tex]\[
\frac{81}{8} \cdot m^2 \cdot n^5
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{81 m^2 n^5}{8}
\][/tex]
Thus, the equivalent expression is [tex]\(\boxed{\frac{81 m^2 n^5}{8}}\)[/tex].
