Solve (1+x/x^{2-xy}-1-y/y^2-xy)*(x+y/xy^2-x^2y)^-1=

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

Evaluate

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Short Solution Steps

Factor x^{2}-xy. Factor y^{2}-xy.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-y\right) and y\left(-x+y\right) is xy\left(-x+y\right). Multiply \frac{1+x}{x\left(x-y\right)} times \frac{-y}{-y}. Multiply \frac{1-y}{y\left(-x+y\right)} times \frac{x}{x}.

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

Since \frac{\left(1+x\right)\left(-1\right)y}{xy\left(-x+y\right)} and \frac{\left(1-y\right)x}{xy\left(-x+y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(1+x\right)\left(-1\right)y-\left(1-y\right)x.

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

Combine like terms in -y-xy-x+yx.

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

To raise \frac{x+y}{xy^{2}-x^{2}y} to a power, raise both numerator and denominator to the power and then divide.

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

Multiply \frac{-y-x}{xy\left(-x+y\right)} times \frac{\left(x+y\right)^{-1}}{\left(xy^{2}-x^{2}y\right)^{-1}} by multiplying numerator times numerator and denominator times denominator.

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

Use the distributive property to multiply -y-x by \left(x+y\right)^{-1}.

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

Use the distributive property to multiply xy by -x+y.

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

Use the distributive property to multiply -yx^{2}+xy^{2} by \left(xy^{2}-x^{2}y\right)^{-1}.

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

Extract the negative sign in -y\left(x+y\right)^{-1}-x\left(x+y\right)^{-1}.

$$\frac{-\left(-\frac{1}{xy^{2}-yx^{2}}yx^{2}+\frac{1}{xy^{2}-yx^{2}}xy^{2}\right)}{-\frac{1}{xy^{2}-yx^{2}}yx^{2}+\frac{1}{xy^{2}-yx^{2}}xy^{2}}$$

Cancel out -\frac{1}{xy^{2}-yx^{2}}yx^{2}+\frac{1}{xy^{2}-yx^{2}}xy^{2} in both numerator and denominator.

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Factor

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