$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}$是什么意思，有数学表达吗

$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}$ 是一个数学表达式，可以进一步化简。

首先，我们将分母有理化，即

$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}} = \frac{(\sqrt{2}+\sqrt{3})(\sqrt{2}+\sqrt{3})}{(\sqrt{2}-\sqrt{3})(\sqrt{2}+\sqrt{3})} + \frac{(\sqrt{2}-\sqrt{3})(\sqrt{2}-\sqrt{3})}{(\sqrt{3}-\sqrt{2})(\sqrt{2}-\sqrt{3})}$

化简后得到：

$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}} = \frac{5\sqrt{2}+7\sqrt{3}}{-1} = -5\sqrt{2}-7\sqrt{3}$

因此，$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}$ 的值等于 $-5\sqrt{2}-7\sqrt{3}$。

相关问题

解释$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}$

$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}$ 是一个数学表达式。我们可以将其分解为两个分式相加的形式，然后进行有理化简。

首先，根据加减分式通分的方法，可以得到：

$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}=\frac{(\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2})}{(\sqrt{2}-\sqrt{3})(\sqrt{3}-\sqrt{2})}+\frac{(\sqrt{2}-\sqrt{3})(\sqrt{2}+\sqrt{3})}{(\sqrt{3}-\sqrt{2})(\sqrt{2}+\sqrt{3})}$

化简后得到：

$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}=\frac{-\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{3}-\sqrt{3}^2+\sqrt{2}^2}{\sqrt{2}^2-\sqrt{3}^2}+\frac{\sqrt{2}^2-\sqrt{3}^2}{\sqrt{3}^2-\sqrt{2}^2}$

化简后得到：

$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}=\frac{\sqrt{2}-\sqrt{3}}{-1}+\frac{\sqrt{2}+\sqrt{3}}{1}=-\sqrt{2}-\sqrt{3}+\sqrt{2}+\sqrt{3}$

化简后得到：

$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}=0$

因此，$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}$ 的值为 $0$。

$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}}$转换成阿拉伯数字表示

首先，我们可以将第一个分式中的分子和分母同时乘以$\sqrt{2}+\sqrt{3}$，将第二个分式中的分子和分母同时乘以$\sqrt{2}-\sqrt{3}$，得到：

$$\begin{aligned} \frac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{2}} &= \frac{(\sqrt{2}+\sqrt{3})(\sqrt{2}+\sqrt{3})}{(\sqrt{2}-\sqrt{3})(\sqrt{2}+\sqrt{3})} + \frac{(\sqrt{2}-\sqrt{3})(\sqrt{2}-\sqrt{3})}{(\sqrt{3}-\sqrt{2})(\sqrt{2}-\sqrt{3})} \\ &= \frac{2\sqrt{2}\cdot\sqrt{3}+5}{-1} + \frac{-2\sqrt{2}\cdot\sqrt{3}+5}{-1} \\ &= -2\sqrt{6}-2\sqrt{2}+10 \\ &= \boxed{10-2\sqrt{6}-2\sqrt{2}}. \end{aligned}$$