Which expression is equivalent to [tex]\left(\frac{4mn}{m^{-2}n^6}\right)^{-2}[/tex]? Assume [tex]m \neq 0, n \neq 0[/tex].

A. [tex]\frac{n^6}{16m^8}[/tex]
B. [tex]\frac{n^{10}}{16m^6}[/tex]
C. [tex]\frac{n^{10}}{8m^8}[/tex]
D. [tex]\frac{4m^3}{n^8}[/tex]



Answer :

Certainly! Let's simplify the given expression step by step.

We start with the expression:
[tex]\[ \left(\frac{4 m n}{m^{-2} n^6}\right)^{-2} \][/tex]

Step 1: Simplify the inner expression:
[tex]\[ \frac{4 m n}{m^{-2} n^6} \][/tex]

First, let's simplify the fraction inside:
[tex]\[ \frac{4 m n}{m^{-2} n^6} = 4 m n \cdot m^2 \cdot \frac{1}{n^6} = 4 m^{1+2} n^{1-6} = 4 m^3 n^{-5} \][/tex]

So, the expression simplifies to:
[tex]\[ (4 m^3 n^{-5})^{-2} \][/tex]

Step 2: Apply the exponent [tex]\(-2\)[/tex] to each term inside the parentheses:
[tex]\[ (4 m^3 n^{-5})^{-2} = 4^{-2} \cdot (m^3)^{-2} \cdot (n^{-5})^{-2} \][/tex]

Step 3: Simplify each term separately:
[tex]\[ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \][/tex]
[tex]\[ (m^3)^{-2} = m^{3 \cdot -2} = m^{-6} \][/tex]
[tex]\[ (n^{-5})^{-2} = n^{-5 \cdot -2} = n^{10} \][/tex]

Combine the simplified terms:
[tex]\[ (4 m^3 n^{-5})^{-2} = \frac{1}{16} \cdot m^{-6} \cdot n^{10} = \frac{n^{10}}{16 m^6} \][/tex]

Therefore, the expression:
[tex]\[ \left(\frac{4 m n}{m^{-2} n^6}\right)^{-2} \][/tex]

is equivalent to:
[tex]\[ \frac{n^{10}}{16 m^6} \][/tex]

Thus, the correct option is:
[tex]\[ \frac{n^{10}}{16 m^6} \][/tex]