回復 7# maymay 的帖子
第 6 題
方程式\(\sqrt{2\sqrt{3}-3}=\sqrt{x\sqrt{3}}-\sqrt{y\sqrt{3}}\)的有理數解\(x,y\)為 。
[解答]
\(\displaystyle\sqrt{2\sqrt{3}-3}\)
\(\displaystyle=\sqrt[4]{3}\cdot\sqrt{2-\sqrt{3}}\)
\(\displaystyle=\sqrt[4]{3}\cdot\frac{\sqrt{4-2\sqrt{3}}}{\sqrt{2}}\)
\(\displaystyle=\frac{\sqrt[4]{3}}{\sqrt{2}}\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(\displaystyle=\frac{\sqrt[4]{3}}{\sqrt{2}}\left(\sqrt{3}-1\right)\)
\(\displaystyle=\sqrt{\frac{3\sqrt{3}}{2}}-\sqrt{\frac{\sqrt{3}}{2}}\)