Solve (x/y-y/x)(y/x+x/y-1)divfrac{x^3+y^3}{x^2y}

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

Evaluate

$$\frac{x}{y}-1$$

Short Solution Steps

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

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

Since \frac{xx}{xy} and \frac{yy}{xy} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xx-yy.

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

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

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

Since \frac{yy}{xy} and \frac{xx}{xy} have the same denominator, add them by adding their numerators.

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

Do the multiplications in yy+xx.

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

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

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

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

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

Multiply \frac{x^{2}-y^{2}}{xy} times \frac{y^{2}+x^{2}-xy}{xy} by multiplying numerator times numerator and denominator times denominator.

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

Divide \frac{\left(x^{2}-y^{2}\right)\left(y^{2}+x^{2}-xy\right)}{xyxy} by \frac{x^{3}+y^{3}}{x^{2}y} by multiplying \frac{\left(x^{2}-y^{2}\right)\left(y^{2}+x^{2}-xy\right)}{xyxy} by the reciprocal of \frac{x^{3}+y^{3}}{x^{2}y}.

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

Cancel out xxy in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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

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

Expand

$$\frac{x}{y}-1$$

Short Solution Steps

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

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

Since \frac{xx}{xy} and \frac{yy}{xy} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xx-yy.

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

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

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

Since \frac{yy}{xy} and \frac{xx}{xy} have the same denominator, add them by adding their numerators.

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

Do the multiplications in yy+xx.

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

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

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

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

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

Multiply \frac{x^{2}-y^{2}}{xy} times \frac{y^{2}+x^{2}-xy}{xy} by multiplying numerator times numerator and denominator times denominator.

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

Divide \frac{\left(x^{2}-y^{2}\right)\left(y^{2}+x^{2}-xy\right)}{xyxy} by \frac{x^{3}+y^{3}}{x^{2}y} by multiplying \frac{\left(x^{2}-y^{2}\right)\left(y^{2}+x^{2}-xy\right)}{xyxy} by the reciprocal of \frac{x^{3}+y^{3}}{x^{2}y}.

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

Cancel out xxy in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

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

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