解決(x-2y/x+2y+frac{x+2y/x-2y)*(1+frac{x^{2}+4y^2}{4xy})}{x^2+4y^2/2xy*(x^2+2xy)} |Microsoft數學求解器

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

評估

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

簡短的解題步驟

若要對運算式相加或相減，請先通分使其分母相同。 x+2y 和 x-2y 的最小公倍式為 \left(x-2y\right)\left(x+2y\right)。 \frac{x-2y}{x+2y} 乘上 \frac{x-2y}{x-2y}。 \frac{x+2y}{x-2y} 乘上 \frac{x+2y}{x+2y}。

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

因為 \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)} 的分母相同，所以將分子相加即可相加這兩個值。

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

計算 \left(x-2y\right)\left(x-2y\right)+\left(x+2y\right)\left(x+2y\right) 的乘法。

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

合併 x^{2}-2xy-2xy+4y^{2}+x^{2}+2xy+2xy+4y^{2} 中的同類項。

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

若要對運算式相加或相減，請先通分使其分母相同。 1 乘上 \frac{4xy}{4xy}。

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

因為 \frac{4xy}{4xy} 和 \frac{x^{2}+4y^{2}}{4xy} 的分母相同，所以將分子相加即可相加這兩個值。

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

\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)} 乘上 \frac{4xy+x^{2}+4y^{2}}{4xy} 的算法: 將分子和分子相乘以及將分母和分母相乘。

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

運算式 \frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right) 為最簡分數。

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

\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} 的算法是將 \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} 的倒數。

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

在分子和分母中同時消去 2xy。

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

因數分解尚未分解的運算式。

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

在分子和分母中同時消去 2\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)。

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

展開運算式。

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

展開

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

簡短的解題步驟

若要對運算式相加或相減，請先通分使其分母相同。 x+2y 和 x-2y 的最小公倍式為 \left(x-2y\right)\left(x+2y\right)。 \frac{x-2y}{x+2y} 乘上 \frac{x-2y}{x-2y}。 \frac{x+2y}{x-2y} 乘上 \frac{x+2y}{x+2y}。

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

因為 \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)} 的分母相同，所以將分子相加即可相加這兩個值。

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

計算 \left(x-2y\right)\left(x-2y\right)+\left(x+2y\right)\left(x+2y\right) 的乘法。

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

合併 x^{2}-2xy-2xy+4y^{2}+x^{2}+2xy+2xy+4y^{2} 中的同類項。

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

若要對運算式相加或相減，請先通分使其分母相同。 1 乘上 \frac{4xy}{4xy}。

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

因為 \frac{4xy}{4xy} 和 \frac{x^{2}+4y^{2}}{4xy} 的分母相同，所以將分子相加即可相加這兩個值。

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

\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)} 乘上 \frac{4xy+x^{2}+4y^{2}}{4xy} 的算法: 將分子和分子相乘以及將分母和分母相乘。

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

運算式 \frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right) 為最簡分數。

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

\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} 的算法是將 \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} 的倒數。

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

在分子和分母中同時消去 2xy。

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

因數分解尚未分解的運算式。

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

在分子和分母中同時消去 2\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)。

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

展開運算式。

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