Calculus Examples

$\sqrt{\frac{x^{2}}{64} + \frac{4}{x} + \frac{1}{2}}$

Step 1

To write $\frac{x^{2}}{64}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\sqrt{\frac{x^{2}}{64} \cdot \frac{x}{x} + \frac{4}{x} + \frac{1}{2}}$

Step 2

To write $\frac{4}{x}$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$\sqrt{\frac{x^{2}}{64} \cdot \frac{x}{x} + \frac{4}{x} \cdot \frac{64}{64} + \frac{1}{2}}$

Step 3

Write each expression with a common denominator of $64 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x^{2}}{64}$ by $\frac{x}{x}$ .

$\sqrt{\frac{x^{2} x}{64 x} + \frac{4}{x} \cdot \frac{64}{64} + \frac{1}{2}}$

Step 3.2

Multiply $\frac{4}{x}$ by $\frac{64}{64}$ .

$\sqrt{\frac{x^{2} x}{64 x} + \frac{4 \cdot 64}{x \cdot 64} + \frac{1}{2}}$

Step 3.3

Reorder the factors of $x \cdot 64$ .

$\sqrt{\frac{x^{2} x}{64 x} + \frac{4 \cdot 64}{64 x} + \frac{1}{2}}$

Step 4

Combine the numerators over the common denominator.

$\sqrt{\frac{x^{2} x + 4 \cdot 64}{64 x} + \frac{1}{2}}$

Step 5

Multiply $4$ by $64$ .

$\sqrt{\frac{x^{2} x + 256}{64 x} + \frac{1}{2}}$

Step 6

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 6.1

Multiply $x^{2}$ by $x$ .

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Step 6.1.1

Raise $x$ to the power of $1$ .

$\sqrt{\frac{x^{2} x^{1} + 256}{64 x} + \frac{1}{2}}$

Step 6.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{\frac{x^{2 + 1} + 256}{64 x} + \frac{1}{2}}$

Step 6.2

Add $2$ and $1$ .

$\sqrt{\frac{x^{3} + 256}{64 x} + \frac{1}{2}}$

Step 7

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{32 x}{32 x}$ .

$\sqrt{\frac{x^{3} + 256}{64 x} + \frac{1}{2} \cdot \frac{32 x}{32 x}}$

Step 8

Write each expression with a common denominator of $64 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{1}{2}$ by $\frac{32 x}{32 x}$ .

$\sqrt{\frac{x^{3} + 256}{64 x} + \frac{32 x}{2 (32 x)}}$

Step 8.2

Multiply $32$ by $2$ .

$\sqrt{\frac{x^{3} + 256}{64 x} + \frac{32 x}{64 x}}$

Step 9

Combine the numerators over the common denominator.

$\sqrt{\frac{x^{3} + 256 + 32 x}{64 x}}$

Step 10

Reorder terms.

$\sqrt{\frac{x^{3} + 32 x + 256}{64 x}}$

Step 11

Rewrite $\frac{x^{3} + 32 x + 256}{64 x}$ as ${(\frac{1}{8})}^{2} \frac{x^{3} + 32 x + 256}{x}$ .

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Step 11.1

Factor the perfect power $1^{2}$ out of $x^{3} + 32 x + 256$ .

$\sqrt{\frac{1^{2} (x^{3} + 32 x + 256)}{64 x}}$

Step 11.2

Factor the perfect power $8^{2}$ out of $64 x$ .

$\sqrt{\frac{1^{2} (x^{3} + 32 x + 256)}{8^{2} x}}$

Step 11.3

Rearrange the fraction $\frac{1^{2} (x^{3} + 32 x + 256)}{8^{2} x}$ .

$\sqrt{{(\frac{1}{8})}^{2} \frac{x^{3} + 32 x + 256}{x}}$

Step 12

Pull terms out from under the radical.

$\frac{1}{8} \sqrt{\frac{x^{3} + 32 x + 256}{x}}$

Step 13

Rewrite $\sqrt{\frac{x^{3} + 32 x + 256}{x}}$ as $\frac{\sqrt{x^{3} + 32 x + 256}}{\sqrt{x}}$ .

$\frac{1}{8} \cdot \frac{\sqrt{x^{3} + 32 x + 256}}{\sqrt{x}}$

Step 14

Combine.

$\frac{1 \sqrt{x^{3} + 32 x + 256}}{8 \sqrt{x}}$

Step 15

Multiply $\sqrt{x^{3} + 32 x + 256}$ by $1$ .

$\frac{\sqrt{x^{3} + 32 x + 256}}{8 \sqrt{x}}$

Step 16

Multiply $\frac{\sqrt{x^{3} + 32 x + 256}}{8 \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{x^{3} + 32 x + 256}}{8 \sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 17

Combine and simplify the denominator.

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Step 17.1

Multiply $\frac{\sqrt{x^{3} + 32 x + 256}}{8 \sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 \sqrt{x} \sqrt{x}}$

Step 17.2

Move $\sqrt{x}$ .

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 (\sqrt{x} \sqrt{x})}$

Step 17.3

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 ({\sqrt{x}}^{1} \sqrt{x})}$

Step 17.4

Raise $\sqrt{x}$ to the power of $1$ .

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 ({\sqrt{x}}^{1} {\sqrt{x}}^{1})}$

Step 17.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 {\sqrt{x}}^{1 + 1}}$

Step 17.6

Add $1$ and $1$ .

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 {\sqrt{x}}^{2}}$

Step 17.7

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

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Step 17.7.1

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

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 {(x^{\frac{1}{2}})}^{2}}$

Step 17.7.2

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

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 x^{\frac{1}{2} \cdot 2}}$

Step 17.7.3

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

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 x^{\frac{2}{2}}}$

Step 17.7.4

Cancel the common factor of $2$ .

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Step 17.7.4.1

Cancel the common factor.

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 x^{\frac{2}{2}}}$

Step 17.7.4.2

Rewrite the expression.

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 x^{1}}$

Step 17.7.5

Simplify.

$\frac{\sqrt{x^{3} + 32 x + 256} \sqrt{x}}{8 x}$

Step 18

Combine using the product rule for radicals.

$\frac{\sqrt{(x^{3} + 32 x + 256) x}}{8 x}$

Step 19

Reorder factors in $\frac{\sqrt{(x^{3} + 32 x + 256) x}}{8 x}$ .

$\frac{\sqrt{x (x^{3} + 32 x + 256)}}{8 x}$