Solve (1/x)^2/(1-(x-1/x)^2)

$$( \frac { 1 } { x } ) ^ { 2 } / ( 1 - ( \frac { x - 1 } { x } ) ^ { 2 } )$$

Evaluate

$$\frac{1}{2x-1}$$

Short Solution Steps

To raise \frac{1}{x} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{x-1}{x} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Since \frac{x^{2}}{x^{2}} and \frac{\left(x-1\right)^{2}}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}-\left(x-1\right)^{2}.

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

Combine like terms in x^{2}-x^{2}+2x-1.

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

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

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

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

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

Calculate 1 to the power of 2 and get 1.

$$\frac{1}{2x-1}$$

Expand

$$\frac{1}{2x-1}$$

Short Solution Steps

To raise \frac{1}{x} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{x-1}{x} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Since \frac{x^{2}}{x^{2}} and \frac{\left(x-1\right)^{2}}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x^{2}-\left(x-1\right)^{2}.

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

Combine like terms in x^{2}-x^{2}+2x-1.

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

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

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

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

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

Calculate 1 to the power of 2 and get 1.

$$\frac{1}{2x-1}$$