Answer:
Given Expression is
[tex](\frac{4mn}{m^{-2}n^{-6}})^{-2}[/tex]
To find the equivalent expression we need the following results.
1. [tex]x^{-a}=\frac{1}{x^a}[/tex]
2. [tex]x^a\times x^b=x^{a+b}[/tex]
3. [tex]\frac{x^a}{x^b}=x^{a-b}[/tex]
4. [tex](x^a)^b=x^{ab}[/tex]
Consider,
[tex](\frac{4mn}{m^{-2}n^{-6}})^{-2}[/tex]
[tex]=(\frac{m^{-2}n^{-6}}{4mn})^{2}[/tex]
[tex]=\frac{1^2}{4^2}\times(\frac{m^{-2}n^{-6}}{mn})^{2}[/tex]
[tex]=\frac{1}{16}\times(m^{-2-1}\times n^{-6-1})^2[/tex]
[tex]=\frac{1}{16}\times(m^{-3}\times n^{-7})^2[/tex]
[tex]=\frac{1}{16}\times(\frac{1}{m^{3}n^{7}})^2[/tex]
[tex]=\frac{1}{16}\times\frac{1^2}{(m^{3}n^{7})^2}[/tex]
[tex]=\frac{1}{16}\times\frac{1}{m^{3\times2}n^{7\times2}}[/tex]
[tex]=\frac{1}{16}\times\frac{1}{m^6n^{14}}[/tex]
[tex]=\frac{1}{16m^6n^{14}}[/tex]
