初中数学题库八年级6.6：关于二次根式的整体思想求值例题解析
解：∵ \(\displaystyle x=\frac{{\sqrt{{n+1}}-\sqrt{n}}}{{\sqrt{{n+1}}+\sqrt{n}}},y=\frac{{\sqrt{{n+1}}+\sqrt{n}}}{{\sqrt{{n+1}}-\sqrt{n}}}\)
∴ \(\displaystyle xy=\frac{{\sqrt{{n+1}}-\sqrt{n}}}{{\sqrt{{n+1}}+\sqrt{n}}}\times \frac{{\sqrt{{n+1}}+\sqrt{n}}}{{\sqrt{{n+1}}-\sqrt{n}}}\)
\(\displaystyle \begin{array}{l}=\frac{{n+1-n}}{{n+1-n}}\\=1\end{array}\)
\(\displaystyle \begin{array}{l}x+y=\frac{{\sqrt{{n+1}}-\sqrt{n}}}{{\sqrt{{n+1}}+\sqrt{n}}}+\frac{{\sqrt{{n+1}}+\sqrt{n}}}{{\sqrt{{n+1}}-\sqrt{n}}}\\=\frac{{{{{\left( {\sqrt{{n+1}}-\sqrt{n}} \right)}}^{2}}+{{{\left( {\sqrt{{n+1}}+\sqrt{n}} \right)}}^{2}}}}{{\left( {\sqrt{{n+1}}+\sqrt{n}} \right)\left( {\sqrt{{n+1}}-\sqrt{n}} \right)}}\\=\frac{{n+1+n-2\sqrt{{{{n}^{2}}+n}}+n+1+n+2\sqrt{{{{n}^{2}}+n}}}}{{n+1-n}}\\=4n+2\end{array}\)
\(\displaystyle \begin{array}{l}2{{x}^{2}}+4xy+2{{y}^{2}}+193xy=1993\\2{{\left( {x+y} \right)}^{2}}+193\times 1=1993\\{{\left( {x+y} \right)}^{2}}=900\\4n+2=\pm \sqrt{{900}}\\4n+2=\pm 30\\{{n}_{1}}=7,{{n}_{2}}-8\end{array}\)
∵ n为自然数
∴ n=-8(舍)
∴ n的值为7
