Solve frac{x^{-3}+y^{-3}}{x^-2-x^-1y^-1+y^-2}

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

Evaluate

$$\frac{x+y}{xy}$$

Short Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

Use the distributive property to multiply y^{3} by -\frac{x^{2}}{y}+x+y^{-2}x^{3}.

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

To multiply powers of the same base, add their exponents. Add 3 and -2 to get 1.

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

Express y^{3}\times \frac{x^{2}}{y} as a single fraction.

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

Cancel out y in both numerator and denominator.

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

Calculate y to the power of 1 and get y.

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

Factor the expressions that are not already factored.

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

Cancel out x^{2}-xy+y^{2} in both numerator and denominator.

$$\frac{x+y}{xy}$$

Expand

$$\frac{x+y}{xy}$$

Short Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

Use the distributive property to multiply y^{3} by -\frac{x^{2}}{y}+x+y^{-2}x^{3}.

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

To multiply powers of the same base, add their exponents. Add 3 and -2 to get 1.

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

Express y^{3}\times \frac{x^{2}}{y} as a single fraction.

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

Cancel out y in both numerator and denominator.

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

Calculate y to the power of 1 and get y.

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

Factor the expressions that are not already factored.

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

Cancel out x^{2}-xy+y^{2} in both numerator and denominator.

$$\frac{x+y}{xy}$$