closest integer value

Question

Find the integer which is closest to the value of

$\frac{1}{\sqrt[6]{5^6+1}- \sqrt[6]{5^6-1}}$

I have tried putting them between n and n+1 and tried manipulating n but I cant seem to find a strategy.

Answer 1

Put $5^{-6}=:p$. Then your number $x$ satisfies $$\eqalign{5x&={1\over(1+p)^{1/6}-(1-p)^{1/6}}={(1+p)^{1/6}+(1-p)^{1/6}\over (1+p)^{1/3}-(1-p)^{1/3}}\cr &={\bigl((1+p)^{1/6}+(1-p)^{1/6}\bigr)\bigl((1+p)^{2/3}+(1-p^2)^{1/3}+(1-p)^{2/3}\bigr)\over2p}\ ,\cr}$$ hence $$x={5^5\over2}\bigl((1+p)^{1/6}+(1-p)^{1/6}\bigr)\bigl((1+p)^{2/3}+(1-p^2)^{1/3}+(1-p)^{2/3}\bigr)\ .\tag{1}$$ Since $p$ is terribly small we conjecture that $$x\approx3\cdot 5^5=9375\ .$$ To make sure we have to develop the RHS of $(1)$ with respect to $p$. Inspection shows that there are no linear terms. We therefore have $$x=9375\>\bigl(1+O(p^2)\bigr)\ ,$$ so that we may safely proclaim $9375$ as end result. For a full proof you would need estimates for the error in the binomial approximation.