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

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

Evaluate

$$4$$

Short Solution Steps

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

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

Since \frac{xx}{x} and \frac{1}{x} have the same denominator, add them by adding their numerators.

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

Do the multiplications in xx+1.

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

To raise \frac{x^{2}+1}{x} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Since \frac{xx}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xx-1.

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

To raise \frac{x^{2}-1}{x} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

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

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

Combine like terms in x^{4}+2x^{2}+1-x^{4}+2x^{2}-1.

$$\frac{4x^{2}}{x^{2}}$$

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

$$4$$

Factor

$$2^{2}$$