Mathematical Enigma - I

When applying the double conjugate this expression will be $\frac{1}{2}(\sqrt{82}+\sqrt{80})(\sqrt[4]{82}+\sqrt[4]{80})$ As $\sqrt{81} = 9; \sqrt[4]{81} = 3$ , the closest approximation of the value will be substituted to the actual value, which will produce $\frac{1}{2}(\sqrt{82}+\sqrt{80})(\sqrt[4]{82}+\sqrt[4]{80})\ \approx \frac{1}{2}(9+9)(3+3) \approx 54$

I used a linear approximation of the roots. This is likely to work because $1 << 3^4$ .

First, divide numerator and denominator by 3: $\frac{1/3}{\sqrt[4]{1+(\tfrac13)^4}-\sqrt[4]{1+(\tfrac13)^4}}.$ Next, apply the approximation $\sqrt[k]{1+x} \approx 1 + \frac x k + O(x^2);$ this results in $\frac{1/3}{(1 + \tfrac14\cdot(\tfrac13)^4) - (1 - \tfrac14\cdot(\tfrac13)^4)} = \frac{1/3}{2\cdot \tfrac14\cdot(\tfrac13)^4} = 2\cdot 3^3 = \boxed{54}.$ How good is this approximation? Note that the remaining terms in the approximation are of the order $O(x^2)$ , that is of the order $(\tfrac1{81})^2 \sim 10^{-4}$ . Thus we expect our solution to be accurate well within 1%. This justifies concluding that 54 is indeed the closest integer.

Direct calculation bears this out: a more exact value obtained by calculator is $53.99819951$ .