17. Let $[x]$ denote the greatest integer not exceeding $x$. Find the last three digits of $\left[(\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2})^{2014}\right]$
(2 marks)
設 $[x]$ 表示不超過 $x$ 的最大整數。求 $\left[(\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2})^{2014}\right]$ 的最後三位數字。
17. 125
17. Let $x=\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2}$. Using the formula $(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)$, we have
$$
\begin{aligned}
x^{3} & =(\sqrt{5}+2)+(\sqrt{5}-2)+3 \sqrt[3]{(\sqrt{5}+2)(\sqrt{5}-2) x} \\
& =2 \sqrt{5}+3 x
\end{aligned}
$$

Rewrite this as $(x-\sqrt{5})\left(x^{2}+\sqrt{5} x+2\right)=0$. Since the quadratic has no real roots, we must have $x=\sqrt{5}$. So $\left[(\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2})^{2014}\right]=\left[\sqrt{5}{ }^{2014}\right]=5^{1007}$. The last 3 digits of the powers of 5 follow the pattern $005,025,125,625,125,625, \ldots$ It follows that the answer is 125 .