Simplify: $\;$ $\left[\dfrac{m - n}{m^{\frac{3}{4}} + m^{\frac{1}{2}} n^{\frac{1}{4}}} - \dfrac{m^{\frac{1}{2}} - n^{\frac{1}{2}}}{m^{\frac{1}{4}} + n^{\frac{1}{4}}}\right] \left[\dfrac{n}{m}\right]^{\frac{-1}{2}}$
$\left[\dfrac{m - n}{m^{\frac{3}{4}} + m^{\frac{1}{2}} n^{\frac{1}{4}}} - \dfrac{m^{\frac{1}{2}} - n^{\frac{1}{2}}}{m^{\frac{1}{4}} + n^{\frac{1}{4}}}\right] \left[\dfrac{n}{m}\right]^{\frac{-1}{2}}$
$= \left[\dfrac{m - n}{m^{\frac{1}{2}} \left(m^{\frac{1}{4}} + n^{\frac{1}{4}}\right)} - \dfrac{m^{\frac{1}{2}} - n^{\frac{1}{2}}}{m^{\frac{1}{4}} + n^{\frac{1}{4}}}\right] \times \dfrac{m^{\frac{1}{2}}}{n^{\frac{1}{2}}}$
$= \left[\dfrac{m - n - m + m^{\frac{1}{2}} n^{\frac{1}{2}}}{m^{\frac{1}{2}} \left(m^{\frac{1}{4}} + n^{\frac{1}{4}}\right)}\right] \times \dfrac{m^{\frac{1}{2}}}{n^{\frac{1}{2}}}$
$= \dfrac{n^{\frac{1}{2}} \left(m^{\frac{1}{2}} - n^{\frac{1}{2}}\right)}{m^{\frac{1}{2}} \left(m^{\frac{1}{4}} + n^{\frac{1}{4}}\right)} \times \dfrac{m^{\frac{1}{2}}}{n^{\frac{1}{2}}}$
$= \dfrac{m^{\frac{1}{2}} - n^{\frac{1}{2}}}{m^{\frac{1}{4}} + n^{\frac{1}{4}}}$
$= \dfrac{\left(m^{\frac{1}{2}} - n^{\frac{1}{2}}\right) \left(m^{\frac{1}{4}} - n^{\frac{1}{4}}\right)}{\left(m^{\frac{1}{4}} + n^{\frac{1}{4}}\right) \left(m^{\frac{1}{4}} - n^{\frac{1}{4}}\right)}$
$= \dfrac{\left(m^{\frac{1}{2}} - n^{\frac{1}{2}}\right) \left(m^{\frac{1}{4}} - n^{\frac{1}{4}}\right)}{m^{\frac{1}{2}} - n^{\frac{1}{2}}}$
$= m^{\frac{1}{4}} - n^{\frac{1}{4}}$