Calculus Examples

\sqrt{\frac{3 x}{8}}

$\sqrt{\frac{3 x}{8}}$

Step 1

Rewrite

\frac{3 x}{8}

$\frac{3 x}{8}$ as

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

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

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Step 1.1

Factor the perfect power

1^{2}

$1^{2}$ out of

3 x

$3 x$ .

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

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

Step 1.2

Factor the perfect power

2^{2}

$2^{2}$ out of

8

$8$ .

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

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

Step 1.3

Rearrange the fraction

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

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

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

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

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

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

Step 2

Pull terms out from under the radical.

\frac{1}{2} \sqrt{\frac{3 x}{2}}

$\frac{1}{2} \sqrt{\frac{3 x}{2}}$

Step 3

Rewrite

\sqrt{\frac{3 x}{2}}

$\sqrt{\frac{3 x}{2}}$ as

\frac{\sqrt{3 x}}{\sqrt{2}}

$\frac{\sqrt{3 x}}{\sqrt{2}}$ .

\frac{1}{2} \cdot \frac{\sqrt{3 x}}{\sqrt{2}}

$\frac{1}{2} \cdot \frac{\sqrt{3 x}}{\sqrt{2}}$

Step 4

Multiply

\frac{\sqrt{3 x}}{\sqrt{2}}

$\frac{\sqrt{3 x}}{\sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply

\frac{\sqrt{3 x}}{\sqrt{2}}

$\frac{\sqrt{3 x}}{\sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{\sqrt{2} \sqrt{2}}

$\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.2

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}

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

Step 5.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}

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

Step 5.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 5.5

Add

1

$1$ and

1

$1$ .

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{2}}

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

Step 5.6

Rewrite

{\sqrt{2}}^{2}

${\sqrt{2}}^{2}$ as

2

$2$ .

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Step 5.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

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

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

Step 5.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}

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

Step 5.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2^{\frac{2}{2}}}

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

Step 5.6.4

Cancel the common factor of

2

$2$ .

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Step 5.6.4.1

Cancel the common factor.

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2^{\frac{2}{2}}}

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

Step 5.6.4.2

Rewrite the expression.

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2^{1}}

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

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2^{1}}

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

Step 5.6.5

Evaluate the exponent.

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2}

$\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2}$

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2}

$\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2}$

\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2}

$\frac{1}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2}$

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

\frac{1}{2} \cdot \frac{\sqrt{3 x \cdot 2}}{2}

$\frac{1}{2} \cdot \frac{\sqrt{3 x \cdot 2}}{2}$

Step 6.2

Multiply

2

$2$ by

3

$3$ .

\frac{1}{2} \cdot \frac{\sqrt{6 x}}{2}

$\frac{1}{2} \cdot \frac{\sqrt{6 x}}{2}$

\frac{1}{2} \cdot \frac{\sqrt{6 x}}{2}

$\frac{1}{2} \cdot \frac{\sqrt{6 x}}{2}$

Step 7

Multiply

\frac{1}{2} \cdot \frac{\sqrt{6 x}}{2}

$\frac{1}{2} \cdot \frac{\sqrt{6 x}}{2}$ .

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Step 7.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

\frac{\sqrt{6 x}}{2}

$\frac{\sqrt{6 x}}{2}$ .

\frac{\sqrt{6 x}}{2 \cdot 2}

$\frac{\sqrt{6 x}}{2 \cdot 2}$

Step 7.2

Multiply

2

$2$ by

2

$2$ .

\frac{\sqrt{6 x}}{4}

$\frac{\sqrt{6 x}}{4}$

\frac{\sqrt{6 x}}{4}

$\frac{\sqrt{6 x}}{4}$

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