Question

10．設x=$\frac{2}{\sqrt{2}+\sqrt{3}-1}$，y=$\frac{2}{\sqrt{2}+\sqrt{3}+1}$，求$\frac{xy}{x+y}$的值．

Answer 1

分析 將x、y的值代入$\frac{xy}{x+y}$=$\frac{1}{\frac{1}{y}+\frac{1}{x}}$計算可得．

解答 解：當x=$\frac{2}{\sqrt{2}+\sqrt{3}-1}$，y=$\frac{2}{\sqrt{2}+\sqrt{3}+1}$時，
原式=$\frac{1}{\frac{1}{y}+\frac{1}{x}}$
=$\frac{1}{\frac{\sqrt{2}+\sqrt{3}+1}{2}+\frac{\sqrt{2}+\sqrt{3}-1}{2}}$
=$\frac{1}{\sqrt{3}+\sqrt{2}}$
=$\frac{\sqrt{3}-\sqrt{2}}{（\sqrt{3}+\sqrt{2}）（\sqrt{3}-\sqrt{2}）}$
=$\sqrt{3}$-$\sqrt{2}$．

點評 本題考查了二次根式的化簡求值，把二次根式化變形成分母為同分母的兩數(shù)相加是解題的關鍵．