Find the integer which is closest to the value of $\frac{1}{\sqrt[6]{5^6+1}-\sqrt[6]{5^6-1}}$
How do you do this? I try to rationalize but that gets to $\frac{\sqrt[6]{5^6+1}+\sqrt[6]{5^6-1}}{\sqrt[3]{5^6+1}-\sqrt[3]{5^6-1}}$ and I really don't know where to proceed from here.
Help would be appreciated.
Partial solution: Remember: $$x^{n + 1} - y^{n + 1} = (x - y) (x^n + x^{n - 1} y + \dotsb + x y^{n - 1} + y^n)$$ use this to rationalize. You get rid of the roots in the denominator, the numerator is a sum of roots, which you'd have to handle next.
But next to $5^6$, 1 is small, check if they make a difference (now you have a sum, easier to handle than the original fraction).
Or you could expand as:
$\begin{align*} \frac{1} {\sqrt[6]{5^6+1} \left( 1 - \sqrt[6]{(5^6 - 1) / (5^6 + 1)} \right)} \end{align*}$
This is just a geometric series, but it looks like the ratio is very near one, so it'll converge slowly, perhaps too slow to get an estimate within 1/2 with a reasonable number of terms.
