Find the closest integer to $\frac{1}{\sqrt[6]{5^6+1}-\sqrt[6]{5^6-1}}$ This appeared in the Area Level of the 19th Philippine Mathematical Olympiad. Electronic calculators were not allowed during the competition of course. The closest I got to was to express it as:
$$
\dfrac{1}{\sqrt[6]{2}(\sqrt[6]{7813}-\sqrt[3]{6}\sqrt[6]{217})}.
$$
The answer using an electronic calculator is 9375. Can anyone show how this can be approximated without the use of electronic calculators?

Consider
$$y=\frac{1}{\sqrt[6]{x^6+1}-\sqrt[6]{x^6-1}}$$ and use series expansions for "large" values of $x$ for each piece and then long division.
You should end with
$$y=3 x^5-\frac{55}{72 x^7}+O\left(\frac{1}{x^{19}}\right)$$
