Precalculus Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

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Step 2.1

Rewrite ${\sqrt{2 x}}^{3}$ as $\sqrt{{(2 x)}^{3}}$ .

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

Step 2.2

Apply the product rule to $2 x$ .

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

Step 2.3

Raise $2$ to the power of $3$ .

$\sqrt{8 x^{3}} + 3 {\sqrt{2 x}}^{2} \frac{1}{x} + 3 \sqrt{2 x} {(\frac{1}{x})}^{2} + {(\frac{1}{x})}^{3}$

Step 2.4

Rewrite $8 x^{3}$ as ${(2 x)}^{2} \cdot (2 x)$ .

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Step 2.4.1

Factor $4$ out of $8$ .

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

Step 2.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.4.3

Factor out $x^{2}$ .

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

Step 2.4.4

Move $2$ .

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

Step 2.4.5

Rewrite $2^{2} x^{2}$ as ${(2 x)}^{2}$ .

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

Step 2.4.6

Add parentheses.

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

Step 2.5

Pull terms out from under the radical.

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

Step 2.6

Rewrite ${\sqrt{2 x}}^{2}$ as $2 x$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x}$ as ${(2 x)}^{\frac{1}{2}}$ .

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

Step 2.6.2

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

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

Step 2.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Simplify.

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

Step 2.7

Cancel the common factor of $x$ .

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Step 2.7.1

Factor $x$ out of $3 (2 x)$ .

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

Step 2.7.2

Cancel the common factor.

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

Step 2.7.3

Rewrite the expression.

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

Step 2.8

Multiply $3$ by $2$ .

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

Step 2.9

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

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

Step 2.10

One to any power is one.

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

Step 2.11

Multiply $3 \sqrt{2 x} \frac{1}{x^{2}}$ .

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Step 2.11.1

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

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

Step 2.11.2

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

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

Step 2.12

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

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

Step 2.13

One to any power is one.

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