Solve (1+2y/x-y)*[(1-2xy/x^{2+xy+y^{2}}):x^3+y^3/x^3-y^3]

$$( 1 + \frac { 2 y } { x - y } ) \cdot [ ( 1 - \frac { 2 x y } { x ^ { 2 } + x y + y ^ { 2 } } ) : \frac { x ^ { 3 } + y ^ { 3 } } { x ^ { 3 } - y ^ { 3 } } ]$$

Evaluate

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Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-y}{x-y}.

$$\left(\frac{x-y}{x-y}+\frac{2y}{x-y}\right)\times \frac{1-\frac{2xy}{x^{2}+xy+y^{2}}}{\frac{x^{3}+y^{3}}{x^{3}-y^{3}}}$$

Since \frac{x-y}{x-y} and \frac{2y}{x-y} have the same denominator, add them by adding their numerators.

$$\frac{x-y+2y}{x-y}\times \frac{1-\frac{2xy}{x^{2}+xy+y^{2}}}{\frac{x^{3}+y^{3}}{x^{3}-y^{3}}}$$

Combine like terms in x-y+2y.

$$\frac{x+y}{x-y}\times \frac{1-\frac{2xy}{x^{2}+xy+y^{2}}}{\frac{x^{3}+y^{3}}{x^{3}-y^{3}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}+xy+y^{2}}{x^{2}+xy+y^{2}}.

$$\frac{x+y}{x-y}\times \frac{\frac{x^{2}+xy+y^{2}}{x^{2}+xy+y^{2}}-\frac{2xy}{x^{2}+xy+y^{2}}}{\frac{x^{3}+y^{3}}{x^{3}-y^{3}}}$$

Since \frac{x^{2}+xy+y^{2}}{x^{2}+xy+y^{2}} and \frac{2xy}{x^{2}+xy+y^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x+y}{x-y}\times \frac{\frac{x^{2}+xy+y^{2}-2xy}{x^{2}+xy+y^{2}}}{\frac{x^{3}+y^{3}}{x^{3}-y^{3}}}$$

Combine like terms in x^{2}+xy+y^{2}-2xy.

$$\frac{x+y}{x-y}\times \frac{\frac{-xy+x^{2}+y^{2}}{x^{2}+xy+y^{2}}}{\frac{x^{3}+y^{3}}{x^{3}-y^{3}}}$$

Divide \frac{-xy+x^{2}+y^{2}}{x^{2}+xy+y^{2}} by \frac{x^{3}+y^{3}}{x^{3}-y^{3}} by multiplying \frac{-xy+x^{2}+y^{2}}{x^{2}+xy+y^{2}} by the reciprocal of \frac{x^{3}+y^{3}}{x^{3}-y^{3}}.

$$\frac{x+y}{x-y}\times \frac{\left(-xy+x^{2}+y^{2}\right)\left(x^{3}-y^{3}\right)}{\left(x^{2}+xy+y^{2}\right)\left(x^{3}+y^{3}\right)}$$

Factor the expressions that are not already factored in \frac{\left(-xy+x^{2}+y^{2}\right)\left(x^{3}-y^{3}\right)}{\left(x^{2}+xy+y^{2}\right)\left(x^{3}+y^{3}\right)}.

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

Cancel out \left(x^{2}-xy+y^{2}\right)\left(x^{2}+xy+y^{2}\right) in both numerator and denominator.

$$\frac{x+y}{x-y}\times \frac{x-y}{x+y}$$

Multiply \frac{x+y}{x-y} times \frac{x-y}{x+y} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.

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Factor

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