jmc

Let $a=5^4+1$ and $b=5^4-1,$ so the given expression becomes $\frac{1}{\sqrt[4]{a}-\sqrt[4]{b}}.$ We can rationalize by multiplying top and bottom by $\sqrt[4]{a}+\sqrt[4]{b},$ and then by $\sqrt{a}+\sqrt{b}$: $$\begin{array}{rl}\frac{1}{\sqrt[4]{a}-\sqrt[4]{b}}& =\frac{\sqrt[4]{a}+\sqrt[4]{b}}{\sqrt{a}-\sqrt{b}}=\frac{\left(\sqrt[4]{a}+\sqrt[4]{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}{a-b}.\end{array}$$We have $a-b=2,$ and both $a$ and $b$ are close to $5^4,$ so we estimate that $$\frac{\left(\sqrt[4]{a}+\sqrt[4]{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}{a-b}\approx \frac{(5+5)(5^2+5^2)}{2}=\boxed{250}.$$(Indeed, the actual value of the expression is $249.9998599\dots .$)