Simplify: $\;$ $\dfrac{a + 10 \sqrt{a} + \sqrt{20} \left(\sqrt[4]{a^3} + 5 \sqrt[4]{a}\right) + 25}{\left(a - 25\right) \left(\sqrt[4]{a^3} - \sqrt{125}\right)^{-1} \left(\sqrt{a} + \sqrt[4]{25a} + 5\right)}$
$\dfrac{a + 10 \sqrt{a} + \sqrt{20} \left(\sqrt[4]{a^3} + 5 \sqrt[4]{a}\right) + 25}{\left(a - 25\right) \left(\sqrt[4]{a^3} - \sqrt{125}\right)^{-1} \left(\sqrt{a} + \sqrt[4]{25a} + 5\right)}$ $\;\;\; \cdots \; (1)$
Consider the expression
${\left(a - 25\right) \left(\sqrt[4]{a^3} - \sqrt{125}\right)^{-1} \left(\sqrt{a} + \sqrt[4]{25a} + 5\right)}$
$= \left(a - 25\right) \left[\left(a^{\frac{1}{4}}\right)^3 - \left(5^{\frac{1}{2}}\right)^3\right]^{-1} \left(a^{\frac{1}{2}} + 25^{\frac{1}{4}} a^{\frac{1}{4}} + 5\right)$
$= \left(a - 25\right) \left[\left(a^{\frac{1}{4}} - 5^{\frac{1}{2}}\right) \left(a^{\frac{1}{2}} + 5^{\frac{1}{2}} a^{\frac{1}{4}} + 5\right)\right]^{-1} \left(a^{\frac{1}{2}} + 25^{\frac{1}{4}} a^{\frac{1}{4}} + 5\right)$
$= \dfrac{\left(a - 25\right) \left(a^{\frac{1}{2}} + 25^{\frac{1}{4}} a^{\frac{1}{4}} + 5\right)}{\left(a^{\frac{1}{4}} - 5^{\frac{1}{2}}\right) \left(a^{\frac{1}{2}} + 5^{\frac{1}{2}} a^{\frac{1}{4}} + 5\right)}$
$= \dfrac{\left(a - 25\right) \left(a^{\frac{1}{2}} + 25^{\frac{1}{4}} a^{\frac{1}{4}} + 5\right)}{\left(a^{\frac{1}{4}} - 5^{\frac{1}{2}}\right) \left(a^{\frac{1}{2}} + 25^{\frac{1}{4}} a^{\frac{1}{4}} + 5\right)}$
$= \dfrac{a - 25}{a^{\frac{1}{4}} - 5^{\frac{1}{2}}}$
$= \dfrac{\left(a^{\frac{1}{2}} + 5\right) \left(a^{\frac{1}{2}} - 5\right)}{a^{\frac{1}{4}} - \sqrt{5}}$ $\;\;\; \cdots \; (2)$
Consider the expression
$a + 10 \sqrt{a} + \sqrt{20} \left(\sqrt[4]{a^3} + 5 \sqrt[4]{a}\right) + 25$
$= a + 10 a^{\frac{1}{2}} + 2 \sqrt{5} a^{\frac{3}{4}} + 10 \sqrt{5} a^{\frac{1}{4}} + 25$
$= \left(a + 10 a^{\frac{1}{2}} + 25\right) + 2 \sqrt{5} a^{\frac{1}{4}} \left(a^{\frac{1}{2}} + 5\right)$
$= \left[\left(a^{\frac{1}{2}}\right)^2 + 2 \times 5 \times a^{\frac{1}{2}} + 5^2\right] + 2 \sqrt{5} a^{\frac{1}{4}} \left(a^{\frac{1}{2}} + 5\right)$
$= \left(a^{\frac{1}{2}} + 5\right)^2 + 2 \sqrt{5} a^{\frac{1}{4}} \left(a^{\frac{1}{2}} + 5\right)$
$= \left(a^{\frac{1}{2}} + 5\right) \left(a^{\frac{1}{2}} + 5 + 2 \sqrt{5} a^{\frac{1}{4}}\right)$
$= \left(a^{\frac{1}{2}} + 5\right) \left[\left(a^{\frac{1}{4}}\right)^2 + 2 \times \sqrt{5} \times a^{\frac{1}{4}} + \left(\sqrt{5}\right)^2\right]$
$= \left(a^{\frac{1}{2}} + 5\right) \left(a^{\frac{1}{4}} + \sqrt{5}\right)^2$ $\;\;\; \cdots \; (3)$
In view of expressions $(2)$ and $(3)$, expression $(1)$ becomes
$\dfrac{\left(a^{\frac{1}{2}} + 5\right) \left(a^{\frac{1}{4}} + \sqrt{5}\right)^2}{\dfrac{\left(a^{\frac{1}{2}} + 5\right) \left(a^{\frac{1}{2}} - 5\right)}{a^{\frac{1}{4}} - \sqrt{5}}}$
$= \dfrac{\left(a^{\frac{1}{4}} + \sqrt{5}\right)^2 \left(a^{\frac{1}{4}} - \sqrt{5}\right)}{a^{\frac{1}{2}} - 5}$
$= \dfrac{\left(a^{\frac{1}{4}} + \sqrt{5}\right) \left(a^{\frac{1}{4}} + \sqrt{5}\right) \left(a^{\frac{1}{4}} - \sqrt{5}\right)}{a^{\frac{1}{2}} - 5}$
$= \dfrac{\left(a^{\frac{1}{4}} + \sqrt{5}\right) \left(a^{\frac{1}{2}} - 5\right)}{a^{\frac{1}{2}} - 5}$
$= \sqrt[4]{a} + \sqrt{5}$