求解 (x-2y/x+2y+frac{x+2y/x-2y)*(1+frac{x^{2}+4y^2}{4xy})}{x^2+4y^2/2xy*(x^2+2xy)}

$$\frac { ( \frac { x - 2 y } { x + 2 y } + \frac { x + 2 y } { x - 2 y } ) \cdot ( 1 + \frac { x ^ { 2 } + 4 y ^ { 2 } } { 4 x y } ) } { \frac { x ^ { 2 } + 4 y ^ { 2 } } { 2 x y } \cdot ( x ^ { 2 } + 2 x y ) }$$

求值

$$\frac{1}{x\left(x-2y\right)}$$

简短求解步骤

若要对表达式执行加法或减法运算，请重写该表达式，使其分母相同。 x+2y 和 x-2y 的最小公倍数是 \left(x-2y\right)\left(x+2y\right)。 求 \frac{x-2y}{x+2y} 与 \frac{x-2y}{x-2y} 的乘积。 求 \frac{x+2y}{x-2y} 与 \frac{x+2y}{x+2y} 的乘积。

$$\frac{\left(\frac{\left(x-2y\right)\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)}+\frac{\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)}\right)\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

由于 \frac{\left(x-2y\right)\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)} 和 \frac{\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)} 具有相同的分母，可通过分子相加来求和。

$$\frac{\frac{\left(x-2y\right)\left(x-2y\right)+\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

完成 \left(x-2y\right)\left(x-2y\right)+\left(x+2y\right)\left(x+2y\right) 中的乘法运算。

$$\frac{\frac{x^{2}-2xy-2xy+4y^{2}+x^{2}+2xy+2xy+4y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

合并 x^{2}-2xy-2xy+4y^{2}+x^{2}+2xy+2xy+4y^{2} 中的项。

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

若要对表达式执行加法或减法运算，请重写该表达式，使其分母相同。 求 1 与 \frac{4xy}{4xy} 的乘积。

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(\frac{4xy}{4xy}+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

由于 \frac{4xy}{4xy} 和 \frac{x^{2}+4y^{2}}{4xy} 具有相同的分母，可通过分子相加来求和。

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\times \frac{4xy+x^{2}+4y^{2}}{4xy}}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)} 乘以 \frac{4xy+x^{2}+4y^{2}}{4xy} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。

$$\frac{\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy}}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

将 \frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right) 化为简分数。

$$\frac{\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy}}{\frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy}}$$

\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy} 除以 \frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy} 的计算方法是用 \frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy} 乘以 \frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy} 的倒数。

$$\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)\times 2xy}{\left(x-2y\right)\left(x+2y\right)\times 4xy\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}$$

消去分子和分母中的 2xy。

$$\frac{\left(2x^{2}+8y^{2}\right)\left(x^{2}+4xy+4y^{2}\right)}{2\left(x-2y\right)\left(x+2y\right)\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}$$

将尚未因式分解的表达式分解因式。

$$\frac{2\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)}{2x\left(x-2y\right)\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)}$$

消去分子和分母中的 2\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)。

$$\frac{1}{x\left(x-2y\right)}$$

展开表达式。

$$\frac{1}{x^{2}-2xy}$$

展开

$$\frac{1}{x\left(x-2y\right)}$$

简短求解步骤

若要对表达式执行加法或减法运算，请重写该表达式，使其分母相同。 x+2y 和 x-2y 的最小公倍数是 \left(x-2y\right)\left(x+2y\right)。 求 \frac{x-2y}{x+2y} 与 \frac{x-2y}{x-2y} 的乘积。 求 \frac{x+2y}{x-2y} 与 \frac{x+2y}{x+2y} 的乘积。

$$\frac{\left(\frac{\left(x-2y\right)\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)}+\frac{\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)}\right)\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

由于 \frac{\left(x-2y\right)\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)} 和 \frac{\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)} 具有相同的分母，可通过分子相加来求和。

$$\frac{\frac{\left(x-2y\right)\left(x-2y\right)+\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

完成 \left(x-2y\right)\left(x-2y\right)+\left(x+2y\right)\left(x+2y\right) 中的乘法运算。

$$\frac{\frac{x^{2}-2xy-2xy+4y^{2}+x^{2}+2xy+2xy+4y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

合并 x^{2}-2xy-2xy+4y^{2}+x^{2}+2xy+2xy+4y^{2} 中的项。

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

若要对表达式执行加法或减法运算，请重写该表达式，使其分母相同。 求 1 与 \frac{4xy}{4xy} 的乘积。

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(\frac{4xy}{4xy}+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

由于 \frac{4xy}{4xy} 和 \frac{x^{2}+4y^{2}}{4xy} 具有相同的分母，可通过分子相加来求和。

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\times \frac{4xy+x^{2}+4y^{2}}{4xy}}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)} 乘以 \frac{4xy+x^{2}+4y^{2}}{4xy} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。

$$\frac{\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy}}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

将 \frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right) 化为简分数。

$$\frac{\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy}}{\frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy}}$$

\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy} 除以 \frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy} 的计算方法是用 \frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy} 乘以 \frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy} 的倒数。

$$\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)\times 2xy}{\left(x-2y\right)\left(x+2y\right)\times 4xy\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}$$

消去分子和分母中的 2xy。

$$\frac{\left(2x^{2}+8y^{2}\right)\left(x^{2}+4xy+4y^{2}\right)}{2\left(x-2y\right)\left(x+2y\right)\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}$$

将尚未因式分解的表达式分解因式。

$$\frac{2\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)}{2x\left(x-2y\right)\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)}$$

消去分子和分母中的 2\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)。

$$\frac{1}{x\left(x-2y\right)}$$

展开表达式。

$$\frac{1}{x^{2}-2xy}$$