$\sqrt[4]n$ has to be slightly bigger than an integer. Thus $n=a^4+1$ (replacing $1$ by a bigger number always results in a greater difference and bigger n, thus is contraproductive). We want $\sqrt[4]{a^4+1} - a < \epsilon$ (here $\epsilon=0.00001$). This means $a^4+1 < (a+\epsilon)^4=a^4+\epsilon(4a^3+6a^2\epsilon+4a\epsilon^2+\epsilon^3)$. Now this requires that $\frac 1{4\epsilon} < a^3$, giving (using my PC ) that $a \geq 30$. A small calculation then shows that $a=30$ works (we don't need to plug it in the first, but rather in the third inequality).