Solve x-1/x/x+frac{1{x}}:x-x^3/x^2+1

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

Evaluate

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

Short Solution Steps

Divide \frac{x-\frac{1}{x}}{x+\frac{1}{x}} by \frac{x-x^{3}}{x^{2}+1} by multiplying \frac{x-\frac{1}{x}}{x+\frac{1}{x}} by the reciprocal of \frac{x-x^{3}}{x^{2}+1}.

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

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

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

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

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

Do the multiplications in xx-1.

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

Express \frac{x^{2}-1}{x}\left(x^{2}+1\right) as a single fraction.

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

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

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

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

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

Do the multiplications in xx+1.

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

Express \frac{x^{2}+1}{x}\left(x-x^{3}\right) as a single fraction.

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

Divide \frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x} by \frac{\left(x^{2}+1\right)\left(x-x^{3}\right)}{x} by multiplying \frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x} by the reciprocal of \frac{\left(x^{2}+1\right)\left(x-x^{3}\right)}{x}.

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

Cancel out x\left(x^{2}+1\right) in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in 1+x.

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

Cancel out \left(x-1\right)\left(-x-1\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

Divide \frac{x-\frac{1}{x}}{x+\frac{1}{x}} by \frac{x-x^{3}}{x^{2}+1} by multiplying \frac{x-\frac{1}{x}}{x+\frac{1}{x}} by the reciprocal of \frac{x-x^{3}}{x^{2}+1}.

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

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

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

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

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

Do the multiplications in xx-1.

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

Express \frac{x^{2}-1}{x}\left(x^{2}+1\right) as a single fraction.

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

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

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

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

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

Do the multiplications in xx+1.

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

Express \frac{x^{2}+1}{x}\left(x-x^{3}\right) as a single fraction.

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

Divide \frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x} by \frac{\left(x^{2}+1\right)\left(x-x^{3}\right)}{x} by multiplying \frac{\left(x^{2}-1\right)\left(x^{2}+1\right)}{x} by the reciprocal of \frac{\left(x^{2}+1\right)\left(x-x^{3}\right)}{x}.

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

Cancel out x\left(x^{2}+1\right) in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in 1+x.

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

Cancel out \left(x-1\right)\left(-x-1\right) in both numerator and denominator.

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