Solve [frac{x^{-1}+y^{-1}}{x^{-1}-y^{-1}}]div[y^-2+x^-2/y^-2-x^-2]

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

Evaluate

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

Short Solution Steps

Divide \frac{x^{-1}+y^{-1}}{x^{-1}-y^{-1}} by \frac{y^{-2}+x^{-2}}{y^{-2}-x^{-2}} by multiplying \frac{x^{-1}+y^{-1}}{x^{-1}-y^{-1}} by the reciprocal of \frac{y^{-2}+x^{-2}}{y^{-2}-x^{-2}}.

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

Factor the expressions that are not already factored.

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

Cancel out y^{-2}\times \frac{1}{x} in both numerator and denominator.

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

Expand the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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}{x} times \frac{y}{y}. Multiply \frac{x}{y} times \frac{x}{x}.

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

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

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

Do the multiplications in \left(-y+x\right)y+xx.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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}.

$$\frac{\frac{-y^{2}x+x^{2}y+x^{3}-y^{3}}{yx^{2}}}{-\frac{yy}{xy}-\frac{xx}{xy}+1+\frac{y^{2}}{x^{2}}}$$

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

$$\frac{\frac{-y^{2}x+x^{2}y+x^{3}-y^{3}}{yx^{2}}}{\frac{-yy-xx}{xy}+1+\frac{y^{2}}{x^{2}}}$$

Do the multiplications in -yy-xx.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored.

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

Extract the negative sign in x-y.

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

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

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

Expand the expression.

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

Expand

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

Short Solution Steps

Divide \frac{x^{-1}+y^{-1}}{x^{-1}-y^{-1}} by \frac{y^{-2}+x^{-2}}{y^{-2}-x^{-2}} by multiplying \frac{x^{-1}+y^{-1}}{x^{-1}-y^{-1}} by the reciprocal of \frac{y^{-2}+x^{-2}}{y^{-2}-x^{-2}}.

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

Factor the expressions that are not already factored.

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

Cancel out y^{-2}\times \frac{1}{x} in both numerator and denominator.

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

Expand the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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}{x} times \frac{y}{y}. Multiply \frac{x}{y} times \frac{x}{x}.

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

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

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

Do the multiplications in \left(-y+x\right)y+xx.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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}.

$$\frac{\frac{-y^{2}x+x^{2}y+x^{3}-y^{3}}{yx^{2}}}{-\frac{yy}{xy}-\frac{xx}{xy}+1+\frac{y^{2}}{x^{2}}}$$

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

$$\frac{\frac{-y^{2}x+x^{2}y+x^{3}-y^{3}}{yx^{2}}}{\frac{-yy-xx}{xy}+1+\frac{y^{2}}{x^{2}}}$$

Do the multiplications in -yy-xx.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored.

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

Extract the negative sign in x-y.

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

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

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

Expand the expression.

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