Solve 1/x^2-1/y^2/frac{1{x^2}-2/xy+frac{1}{y^2}}

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and y^{2} is x^{2}y^{2}. Multiply \frac{1}{x^{2}} times \frac{y^{2}}{y^{2}}. Multiply \frac{1}{y^{2}} times \frac{x^{2}}{x^{2}}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and xy is yx^{2}. Multiply \frac{1}{x^{2}} times \frac{y}{y}. Multiply \frac{2}{xy} times \frac{x}{x}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of yx^{2} and y^{2} is x^{2}y^{2}. Multiply \frac{y-2x}{yx^{2}} times \frac{y}{y}. Multiply \frac{1}{y^{2}} times \frac{x^{2}}{x^{2}}.

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored.

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

Cancel out -x+y in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and y^{2} is x^{2}y^{2}. Multiply \frac{1}{x^{2}} times \frac{y^{2}}{y^{2}}. Multiply \frac{1}{y^{2}} times \frac{x^{2}}{x^{2}}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and xy is yx^{2}. Multiply \frac{1}{x^{2}} times \frac{y}{y}. Multiply \frac{2}{xy} times \frac{x}{x}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of yx^{2} and y^{2} is x^{2}y^{2}. Multiply \frac{y-2x}{yx^{2}} times \frac{y}{y}. Multiply \frac{1}{y^{2}} times \frac{x^{2}}{x^{2}}.

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored.

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

Cancel out -x+y in both numerator and denominator.

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