Simplify (x+1/x)^3

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Cancel the common factor of $x$ .

Factor $x$ out of $3 x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Apply the product rule to $\frac{1}{x}$ .

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

Cancel the common factor of $x$ .

Factor $x$ out of $3 x$ .

$x^{3} + 3 x + x \cdot 3 \frac{1^{2}}{x^{2}} + {(\frac{1}{x})}^{3}$

Factor $x$ out of $x^{2}$ .

$x^{3} + 3 x + x \cdot 3 \frac{1^{2}}{x \cdot x} + {(\frac{1}{x})}^{3}$

Cancel the common factor.

$x^{3} + 3 x + x \cdot 3 \frac{1^{2}}{x \cdot x} + {(\frac{1}{x})}^{3}$

Rewrite the expression.

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

Combine $3$ and $\frac{1^{2}}{x}$ .

$x^{3} + 3 x + \frac{3 \cdot 1^{2}}{x} + {(\frac{1}{x})}^{3}$

One to any power is one.

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

Multiply $3$ by $1$ .

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

Apply the product rule to $\frac{1}{x}$ .

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

One to any power is one.

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