Which expression is equivalent to [tex]\frac{\left(3 m^{-1} n^2\right)^4}{\left(2 m^{-2} n\right)^3}[/tex]? Assume [tex]m \neq 0[/tex], [tex]n \neq 0[/tex].

A. [tex]2 m^2 n^5[/tex]

B. [tex]\frac{81 m^2 n^5}{8}[/tex]

C. [tex]2 m^2 n^2[/tex]

D. [tex]\frac{81 m^2 n^2}{8}[/tex]



Answer :

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]