Solve (x+1/x)^3-x^3+1/x^3+3(x+1/x)

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

Evaluate

$$6x+\frac{6}{x}+\frac{2}{x^{3}}$$

Short Solution Steps

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

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

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

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

Do the multiplications in xx+1.

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

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

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

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

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

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

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

Do the multiplications in xx+1.

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

Express 3\times \frac{x^{2}+1}{x} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{3} times \frac{x^{3}}{x^{3}}.

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

Since \frac{\left(x^{2}+1\right)^{3}}{x^{3}} and \frac{x^{3}x^{3}}{x^{3}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{2}+1\right)^{3}-x^{3}x^{3}.

$$\frac{\left(x^{2}\right)^{3}+3\left(x^{2}\right)^{2}+3x^{2}\times 1^{2}+1^{3}-x^{6}}{x^{3}}+\frac{1}{x^{3}}+\frac{3\left(x^{2}+1\right)}{x}$$

Combine like terms in \left(x^{2}\right)^{3}+3\left(x^{2}\right)^{2}+3x^{2}\times 1^{2}+1^{3}-x^{6}.

$$\frac{3x^{4}+3x^{2}+1}{x^{3}}+\frac{1}{x^{3}}+\frac{3\left(x^{2}+1\right)}{x}$$

Since \frac{3x^{4}+3x^{2}+1}{x^{3}} and \frac{1}{x^{3}} have the same denominator, add them by adding their numerators.

$$\frac{3x^{4}+3x^{2}+1+1}{x^{3}}+\frac{3\left(x^{2}+1\right)}{x}$$

Combine like terms in 3x^{4}+3x^{2}+1+1.

$$\frac{3x^{4}+3x^{2}+2}{x^{3}}+\frac{3\left(x^{2}+1\right)}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{3} and x is x^{3}. Multiply \frac{3\left(x^{2}+1\right)}{x} times \frac{x^{2}}{x^{2}}.

$$\frac{3x^{4}+3x^{2}+2}{x^{3}}+\frac{3\left(x^{2}+1\right)x^{2}}{x^{3}}$$

Since \frac{3x^{4}+3x^{2}+2}{x^{3}} and \frac{3\left(x^{2}+1\right)x^{2}}{x^{3}} have the same denominator, add them by adding their numerators.

$$\frac{3x^{4}+3x^{2}+2+3\left(x^{2}+1\right)x^{2}}{x^{3}}$$

Do the multiplications in 3x^{4}+3x^{2}+2+3\left(x^{2}+1\right)x^{2}.

$$\frac{3x^{4}+3x^{2}+2+3x^{4}+3x^{2}}{x^{3}}$$

Combine like terms in 3x^{4}+3x^{2}+2+3x^{4}+3x^{2}.

$$\frac{6x^{4}+6x^{2}+2}{x^{3}}$$

Expand

$$6x+\frac{6}{x}+\frac{2}{x^{3}}$$

Short Solution Steps

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

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

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

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

Do the multiplications in xx+1.

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

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

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

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

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

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

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

Do the multiplications in xx+1.

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

Express 3\times \frac{x^{2}+1}{x} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{3} times \frac{x^{3}}{x^{3}}.

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

Since \frac{\left(x^{2}+1\right)^{3}}{x^{3}} and \frac{x^{3}x^{3}}{x^{3}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{2}+1\right)^{3}-x^{3}x^{3}.

$$\frac{\left(x^{2}\right)^{3}+3\left(x^{2}\right)^{2}+3x^{2}\times 1^{2}+1^{3}-x^{6}}{x^{3}}+\frac{1}{x^{3}}+\frac{3\left(x^{2}+1\right)}{x}$$

Combine like terms in \left(x^{2}\right)^{3}+3\left(x^{2}\right)^{2}+3x^{2}\times 1^{2}+1^{3}-x^{6}.

$$\frac{3x^{4}+3x^{2}+1}{x^{3}}+\frac{1}{x^{3}}+\frac{3\left(x^{2}+1\right)}{x}$$

Since \frac{3x^{4}+3x^{2}+1}{x^{3}} and \frac{1}{x^{3}} have the same denominator, add them by adding their numerators.

$$\frac{3x^{4}+3x^{2}+1+1}{x^{3}}+\frac{3\left(x^{2}+1\right)}{x}$$

Combine like terms in 3x^{4}+3x^{2}+1+1.

$$\frac{3x^{4}+3x^{2}+2}{x^{3}}+\frac{3\left(x^{2}+1\right)}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{3} and x is x^{3}. Multiply \frac{3\left(x^{2}+1\right)}{x} times \frac{x^{2}}{x^{2}}.

$$\frac{3x^{4}+3x^{2}+2}{x^{3}}+\frac{3\left(x^{2}+1\right)x^{2}}{x^{3}}$$

Since \frac{3x^{4}+3x^{2}+2}{x^{3}} and \frac{3\left(x^{2}+1\right)x^{2}}{x^{3}} have the same denominator, add them by adding their numerators.

$$\frac{3x^{4}+3x^{2}+2+3\left(x^{2}+1\right)x^{2}}{x^{3}}$$

Do the multiplications in 3x^{4}+3x^{2}+2+3\left(x^{2}+1\right)x^{2}.

$$\frac{3x^{4}+3x^{2}+2+3x^{4}+3x^{2}}{x^{3}}$$

Combine like terms in 3x^{4}+3x^{2}+2+3x^{4}+3x^{2}.

$$\frac{6x^{4}+6x^{2}+2}{x^{3}}$$