Calculus Examples

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

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

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

Step 1.2

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.3.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.3.4

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

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

Step 1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{3}}{2 \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}}$

Step 1.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 1.3.6.2

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

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

Step 1.3.6.3

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

$\frac{\sqrt{3}}{2 \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{3}}{2 \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 1.3.6.4.2

Rewrite the expression.

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

Step 1.3.6.5

Evaluate the exponent.

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

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

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

Step 1.4.2

Multiply $2$ by $3$ .

$\frac{\sqrt{3}}{2 \frac{\sqrt{6}}{3}}$

Step 2

Combine $2$ and $\frac{\sqrt{6}}{3}$ .

$\frac{\sqrt{3}}{\frac{2 \sqrt{6}}{3}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{3} \frac{3}{2 \sqrt{6}}$

Step 4

Multiply $\frac{3}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sqrt{3} (\frac{3}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{3}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sqrt{3} \frac{3 \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Step 5.2

Move $\sqrt{6}$ .

$\sqrt{3} \frac{3 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 5.3

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 5.4

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 5.5

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

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

Step 5.6

Add $1$ and $1$ .

$\sqrt{3} \frac{3 \sqrt{6}}{2 {\sqrt{6}}^{2}}$

Step 5.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 5.7.2

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

$\sqrt{3} \frac{3 \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 5.7.3

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

$\sqrt{3} \frac{3 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

$\sqrt{3} \frac{3 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 5.7.4.2

Rewrite the expression.

$\sqrt{3} \frac{3 \sqrt{6}}{2 \cdot 6^{1}}$

Step 5.7.5

Evaluate the exponent.

$\sqrt{3} \frac{3 \sqrt{6}}{2 \cdot 6}$

Step 6

Cancel the common factor of $3$ and $6$ .

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Step 6.1

Factor $3$ out of $3 \sqrt{6}$ .

$\sqrt{3} \frac{3 (\sqrt{6})}{2 \cdot 6}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $3$ out of $2 \cdot 6$ .

$\sqrt{3} \frac{3 (\sqrt{6})}{3 (2 \cdot 2)}$

Step 6.2.2

Cancel the common factor.

$\sqrt{3} \frac{3 \sqrt{6}}{3 (2 \cdot 2)}$

Step 6.2.3

Rewrite the expression.

$\sqrt{3} \frac{\sqrt{6}}{2 \cdot 2}$

Step 7

Multiply $2$ by $2$ .

$\sqrt{3} \frac{\sqrt{6}}{4}$

Step 8

Multiply $\sqrt{3} \frac{\sqrt{6}}{4}$ .

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Step 8.1

Combine $\sqrt{3}$ and $\frac{\sqrt{6}}{4}$ .

$\frac{\sqrt{3} \sqrt{6}}{4}$

Step 8.2

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 6}}{4}$

Step 8.3

Multiply $3$ by $6$ .

$\frac{\sqrt{18}}{4}$

Step 9

Simplify the numerator.

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Step 9.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 9.1.1

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)}}{4}$

Step 9.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 2}}{4}$

Step 9.2

Pull terms out from under the radical.

$\frac{3 \sqrt{2}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{2}}{4}$

Decimal Form:

$1.06066017 \dots$