Trigonometry Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 3

Rewrite $- 1$ as $i^{2}$ .

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Evaluate the exponent.

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

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

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

Step 8.2

Multiply $3$ by $2$ .

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

Step 9

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

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