Trigonometry Examples

$\sqrt{\frac{336}{\sqrt{3}}}$

Step 1

Multiply $\frac{336}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt{\frac{336}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{336}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt{\frac{336 \sqrt{3}}{\sqrt{3} \sqrt{3}}}$

Step 2.2

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

$\sqrt{\frac{336 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}$

Step 2.3

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

$\sqrt{\frac{336 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}$

Step 2.4

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

$\sqrt{\frac{336 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}}$

Step 2.5

Add $1$ and $1$ .

$\sqrt{\frac{336 \sqrt{3}}{{\sqrt{3}}^{2}}}$

Step 2.6

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

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Step 2.6.1

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

$\sqrt{\frac{336 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}}$

Step 2.6.2

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

$\sqrt{\frac{336 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}$

Step 2.6.3

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

$\sqrt{\frac{336 \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{336 \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{336 \sqrt{3}}{3^{1}}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{336 \sqrt{3}}{3}}$

Step 3

Cancel the common factor of $336$ and $3$ .

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Step 3.1

Factor $3$ out of $336 \sqrt{3}$ .

$\sqrt{\frac{3 (112 \sqrt{3})}{3}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $3$ out of $3$ .

$\sqrt{\frac{3 (112 \sqrt{3})}{3 (1)}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{3 (112 \sqrt{3})}{3 \cdot 1}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{112 \sqrt{3}}{1}}$

Step 3.2.4

Divide $112 \sqrt{3}$ by $1$ .

$\sqrt{112 \sqrt{3}}$

Step 4

Rewrite $112 \sqrt{3}$ as ${(2^{2})}^{2} \cdot (7 \sqrt{3})$ .

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Step 4.1

Factor $16$ out of $112$ .

$\sqrt{16 (7) \sqrt{3}}$

Step 4.2

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} \cdot 7 \sqrt{3}}$

Step 4.3

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

$\sqrt{{(2^{2})}^{2} \cdot 7 \sqrt{3}}$

Step 4.4

Add parentheses.

$\sqrt{{(2^{2})}^{2} \cdot (7 \sqrt{3})}$

Step 5

Pull terms out from under the radical.

$2^{2} \sqrt{7 \sqrt{3}}$

Step 6

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

$4 \sqrt{7 \sqrt{3}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{7 \sqrt{3}}$

Decimal Form:

$13.92801818 \dots$