Solve (x^-1+y^-1)/x^-2*y+x*y^-2

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Expand the expression.

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

Express \frac{1}{y}x^{2} as a single fraction.

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

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

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

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

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

Express \frac{\frac{xy+x^{2}}{y}}{y+y^{-2}x^{3}} as a single fraction.

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

Factor the expressions that are not already factored.

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

Cancel out x+y in both numerator and denominator.

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

Expand the expression.

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

Express \frac{1}{y}x^{2} as a single fraction.

$$\frac{x}{y-x+\frac{x^{2}}{y}}$$

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

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

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

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

Do the multiplications in \left(y-x\right)y+x^{2}.

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

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

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Expand the expression.

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

Express \frac{1}{y}x^{2} as a single fraction.

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

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

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

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

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

Express \frac{\frac{xy+x^{2}}{y}}{y+y^{-2}x^{3}} as a single fraction.

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

Factor the expressions that are not already factored.

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

Cancel out x+y in both numerator and denominator.

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

Expand the expression.

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

Express \frac{1}{y}x^{2} as a single fraction.

$$\frac{x}{y-x+\frac{x^{2}}{y}}$$

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

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

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

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

Do the multiplications in \left(y-x\right)y+x^{2}.

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

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

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