Expand the Trigonometric Expression (x^2+1/x)^3

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

Multiply the exponents in ${(x^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $2$ by $3$ .

$x^{6} + 3 {(x^{2})}^{2} \frac{1}{x} + 3 x^{2} {(\frac{1}{x})}^{2} + {(\frac{1}{x})}^{3}$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{6} + 3 x^{2 \cdot 2} \frac{1}{x} + 3 x^{2} {(\frac{1}{x})}^{2} + {(\frac{1}{x})}^{3}$

Multiply $2$ by $2$ .

$x^{6} + 3 x^{4} \frac{1}{x} + 3 x^{2} {(\frac{1}{x})}^{2} + {(\frac{1}{x})}^{3}$

Cancel the common factor of $x$ .

Factor $x$ out of $3 x^{4}$ .

$x^{6} + x (3 x^{3}) \frac{1}{x} + 3 x^{2} {(\frac{1}{x})}^{2} + {(\frac{1}{x})}^{3}$

Cancel the common factor.

$x^{6} + x (3 x^{3}) \frac{1}{x} + 3 x^{2} {(\frac{1}{x})}^{2} + {(\frac{1}{x})}^{3}$

Rewrite the expression.

$x^{6} + 3 x^{3} + 3 x^{2} {(\frac{1}{x})}^{2} + {(\frac{1}{x})}^{3}$

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

$x^{6} + 3 x^{3} + 3 x^{2} \frac{1^{2}}{x^{2}} + {(\frac{1}{x})}^{3}$

Cancel the common factor of $x^{2}$ .

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

$x^{6} + 3 x^{3} + x^{2} \cdot 3 \frac{1^{2}}{x^{2}} + {(\frac{1}{x})}^{3}$

Cancel the common factor.

$x^{6} + 3 x^{3} + x^{2} \cdot 3 \frac{1^{2}}{x^{2}} + {(\frac{1}{x})}^{3}$

Rewrite the expression.

$x^{6} + 3 x^{3} + 3 \cdot 1^{2} + {(\frac{1}{x})}^{3}$

One to any power is one.

$x^{6} + 3 x^{3} + 3 \cdot 1 + {(\frac{1}{x})}^{3}$

Multiply $3$ by $1$ .

$x^{6} + 3 x^{3} + 3 + {(\frac{1}{x})}^{3}$

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

$x^{6} + 3 x^{3} + 3 + \frac{1^{3}}{x^{3}}$

One to any power is one.

$x^{6} + 3 x^{3} + 3 + \frac{1}{x^{3}}$