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