Solve (x^-1+y^-1/x^-3+y^-3)

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Express \frac{1}{y}x^{3} as a single fraction.

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

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

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

Since \frac{x^{2}y}{y} and \frac{x^{3}}{y} have the same denominator, add them by adding their numerators.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{3}}{y^{3}}.

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

Since \frac{y^{3}}{y^{3}} and \frac{x^{3}}{y^{3}} have the same denominator, add them by adding their numerators.

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

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

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+y in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Express \frac{1}{y}x^{3} as a single fraction.

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

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

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

Since \frac{x^{2}y}{y} and \frac{x^{3}}{y} have the same denominator, add them by adding their numerators.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{3}}{y^{3}}.

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

Since \frac{y^{3}}{y^{3}} and \frac{x^{3}}{y^{3}} have the same denominator, add them by adding their numerators.

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

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

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+y in both numerator and denominator.

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