Solve x^-2+y^-2/x^-2/frac{x^-2{y^-2+{x}^-4}{{x}^-1}}

$$\frac{ \frac{ { x }^{ -2 } + { y }^{ -2 } }{ { x }^{ -2 } } }{ \frac{ { x }^{ -2 } { y }^{ -2 } + { x }^{ -4 } }{ { x }^{ -1 } } }$$

Evaluate

$$x^{3}$$

Short Solution Steps

Divide \frac{x^{-2}+y^{-2}}{x^{-2}} by \frac{x^{-2}y^{-2}+x^{-4}}{x^{-1}} by multiplying \frac{x^{-2}+y^{-2}}{x^{-2}} by the reciprocal of \frac{x^{-2}y^{-2}+x^{-4}}{x^{-1}}.

$$\frac{\left(x^{-2}+y^{-2}\right)x^{-1}}{x^{-2}\left(x^{-2}y^{-2}+x^{-4}\right)}$$

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

$$\frac{\left(x^{-2}+y^{-2}\right)x^{1}}{x^{-2}y^{-2}+x^{-4}}$$

Calculate x to the power of 1 and get x.

$$\frac{\left(x^{-2}+y^{-2}\right)x}{x^{-2}y^{-2}+x^{-4}}$$

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

$$\frac{x^{3}+y^{-2}x^{5}}{1+\left(\frac{1}{y}x\right)^{2}}$$

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

$$\frac{x^{3}+y^{-2}x^{5}}{1+\left(\frac{x}{y}\right)^{2}}$$

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

$$\frac{x^{3}+y^{-2}x^{5}}{1+\frac{x^{2}}{y^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{2}}{y^{2}}.

$$\frac{x^{3}+y^{-2}x^{5}}{\frac{y^{2}}{y^{2}}+\frac{x^{2}}{y^{2}}}$$

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

$$\frac{x^{3}+y^{-2}x^{5}}{\frac{y^{2}+x^{2}}{y^{2}}}$$

Divide x^{3}+y^{-2}x^{5} by \frac{y^{2}+x^{2}}{y^{2}} by multiplying x^{3}+y^{-2}x^{5} by the reciprocal of \frac{y^{2}+x^{2}}{y^{2}}.

$$\frac{\left(x^{3}+y^{-2}x^{5}\right)y^{2}}{y^{2}+x^{2}}$$

Factor the expressions that are not already factored.

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

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

$$y^{-2}y^{2}x^{3}$$

Expand the expression.

$$x^{3}$$

Expand

$$x^{3}$$

Short Solution Steps

Divide \frac{x^{-2}+y^{-2}}{x^{-2}} by \frac{x^{-2}y^{-2}+x^{-4}}{x^{-1}} by multiplying \frac{x^{-2}+y^{-2}}{x^{-2}} by the reciprocal of \frac{x^{-2}y^{-2}+x^{-4}}{x^{-1}}.

$$\frac{\left(x^{-2}+y^{-2}\right)x^{-1}}{x^{-2}\left(x^{-2}y^{-2}+x^{-4}\right)}$$

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

$$\frac{\left(x^{-2}+y^{-2}\right)x^{1}}{x^{-2}y^{-2}+x^{-4}}$$

Calculate x to the power of 1 and get x.

$$\frac{\left(x^{-2}+y^{-2}\right)x}{x^{-2}y^{-2}+x^{-4}}$$

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

$$\frac{x^{3}+y^{-2}x^{5}}{1+\left(\frac{1}{y}x\right)^{2}}$$

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

$$\frac{x^{3}+y^{-2}x^{5}}{1+\left(\frac{x}{y}\right)^{2}}$$

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

$$\frac{x^{3}+y^{-2}x^{5}}{1+\frac{x^{2}}{y^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{2}}{y^{2}}.

$$\frac{x^{3}+y^{-2}x^{5}}{\frac{y^{2}}{y^{2}}+\frac{x^{2}}{y^{2}}}$$

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

$$\frac{x^{3}+y^{-2}x^{5}}{\frac{y^{2}+x^{2}}{y^{2}}}$$

Divide x^{3}+y^{-2}x^{5} by \frac{y^{2}+x^{2}}{y^{2}} by multiplying x^{3}+y^{-2}x^{5} by the reciprocal of \frac{y^{2}+x^{2}}{y^{2}}.

$$\frac{\left(x^{3}+y^{-2}x^{5}\right)y^{2}}{y^{2}+x^{2}}$$

Factor the expressions that are not already factored.

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

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

$$y^{-2}y^{2}x^{3}$$

Expand the expression.

$$x^{3}$$