Let $x=\left(5^{4}+1\right)^{1 / 4}$ and $y=\left(5^{4}-1\right)^{1 / 4}$. Note that $x$ and $y$ are both approximately 5. We have $$\frac{1}{x-y} =\frac{(x+y)\left(x^{2}+y^{2}\right)}{(x-y)(x+y)\left(x^{2}+y^{2}\right)}=\frac{(x+y)\left(x^{2}+y^{2}\right)}{x^{4}-y^{4}} =\frac{(x+y)\left(x^{2}+y^{2}\right)}{2} \approx \frac{(5+5)\left(5^{2}+5^{2}\right)}{2}=250$$