Question

Which expression is equivalent to [4mn/m⁻2n⁻6}⁻2? Assume m and n do not equal zero

Answer 1

The answer would be {(n^5)^2}/{(4*m^3)^2} = { n^{10} }/{16*m^6}

Answer 2

Answer:

Given Expression is

$(\frac{4mn}{m^{-2}n^{-6}})^{-2}$

To find the equivalent expression we need the following results.

1. $x^{-a}=\frac{1}{x^a}$

2. $x^a\times x^b=x^{a+b}$

3.  $\frac{x^a}{x^b}=x^{a-b}$

4. $(x^a)^b=x^{ab}$

Consider,

$(\frac{4mn}{m^{-2}n^{-6}})^{-2}$

$=(\frac{m^{-2}n^{-6}}{4mn})^{2}$

$=\frac{1^2}{4^2}\times(\frac{m^{-2}n^{-6}}{mn})^{2}$

$=\frac{1}{16}\times(m^{-2-1}\times n^{-6-1})^2$

$=\frac{1}{16}\times(m^{-3}\times n^{-7})^2$

$=\frac{1}{16}\times(\frac{1}{m^{3}n^{7}})^2$

$=\frac{1}{16}\times\frac{1^2}{(m^{3}n^{7})^2}$

$=\frac{1}{16}\times\frac{1}{m^{3\times2}n^{7\times2}}$

$=\frac{1}{16}\times\frac{1}{m^6n^{14}}$

$=\frac{1}{16m^6n^{14}}$