Trigonometry Examples

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

Step 1

Raise $3$ to the power of $2$ .

$\sqrt{- \frac{\sqrt{3}}{9} + \frac{\sqrt{6}}{3^{2}}}$

Step 2

To write $- \frac{\sqrt{3}}{9}$ as a fraction with a common denominator, multiply by $\frac{3^{2}}{3^{2}}$ .

$\sqrt{- \frac{\sqrt{3}}{9} \cdot \frac{3^{2}}{3^{2}} + \frac{\sqrt{6}}{3^{2}}}$

Step 3

To write $\frac{\sqrt{6}}{3^{2}}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{- \frac{\sqrt{3}}{9} \cdot \frac{3^{2}}{3^{2}} + \frac{\sqrt{6}}{3^{2}} \cdot \frac{9}{9}}$

Step 4

Write each expression with a common denominator of $9 \cdot 3^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{\sqrt{3}}{9}$ by $\frac{3^{2}}{3^{2}}$ .

$\sqrt{- \frac{\sqrt{3} \cdot 3^{2}}{9 \cdot 3^{2}} + \frac{\sqrt{6}}{3^{2}} \cdot \frac{9}{9}}$

Step 4.2

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

$\sqrt{- \frac{\sqrt{3} \cdot 3^{2}}{3^{2} \cdot 3^{2}} + \frac{\sqrt{6}}{3^{2}} \cdot \frac{9}{9}}$

Step 4.3

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

$\sqrt{- \frac{\sqrt{3} \cdot 3^{2}}{3^{2 + 2}} + \frac{\sqrt{6}}{3^{2}} \cdot \frac{9}{9}}$

Step 4.4

Add $2$ and $2$ .

$\sqrt{- \frac{\sqrt{3} \cdot 3^{2}}{3^{4}} + \frac{\sqrt{6}}{3^{2}} \cdot \frac{9}{9}}$

Step 4.5

Multiply $\frac{\sqrt{6}}{3^{2}}$ by $\frac{9}{9}$ .

$\sqrt{- \frac{\sqrt{3} \cdot 3^{2}}{3^{4}} + \frac{\sqrt{6} \cdot 9}{3^{2} \cdot 9}}$

Step 4.6

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

$\sqrt{- \frac{\sqrt{3} \cdot 3^{2}}{3^{4}} + \frac{\sqrt{6} \cdot 9}{3^{2} \cdot 3^{2}}}$

Step 4.7

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

$\sqrt{- \frac{\sqrt{3} \cdot 3^{2}}{3^{4}} + \frac{\sqrt{6} \cdot 9}{3^{2 + 2}}}$

Step 4.8

Add $2$ and $2$ .

$\sqrt{- \frac{\sqrt{3} \cdot 3^{2}}{3^{4}} + \frac{\sqrt{6} \cdot 9}{3^{4}}}$

Step 5

Combine the numerators over the common denominator.

$\sqrt{\frac{- \sqrt{3} \cdot 3^{2} + \sqrt{6} \cdot 9}{3^{4}}}$

Step 6

Rewrite $\frac{- \sqrt{3} \cdot 3^{2} + \sqrt{6} \cdot 9}{3^{4}}$ in a factored form.

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Step 6.1

Raise $3$ to the power of $2$ .

$\sqrt{\frac{- \sqrt{3} \cdot 9 + \sqrt{6} \cdot 9}{3^{4}}}$

Step 6.2

Multiply $9$ by $- 1$ .

$\sqrt{\frac{- 9 \sqrt{3} + \sqrt{6} \cdot 9}{3^{4}}}$

Step 6.3

Move $9$ to the left of $\sqrt{6}$ .

$\sqrt{\frac{- 9 \sqrt{3} + 9 \sqrt{6}}{3^{4}}}$

Step 7

Raise $3$ to the power of $4$ .

$\sqrt{\frac{- 9 \sqrt{3} + 9 \sqrt{6}}{81}}$

Step 8

Cancel the common factor of $- 9 \sqrt{3} + 9 \sqrt{6}$ and $81$ .

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Step 8.1

Factor $9$ out of $- 9 \sqrt{3}$ .

$\sqrt{\frac{9 (- \sqrt{3}) + 9 \sqrt{6}}{81}}$

Step 8.2

Factor $9$ out of $9 \sqrt{6}$ .

$\sqrt{\frac{9 (- \sqrt{3}) + 9 (\sqrt{6})}{81}}$

Step 8.3

Factor $9$ out of $9 (- \sqrt{3}) + 9 (\sqrt{6})$ .

$\sqrt{\frac{9 (- \sqrt{3} + \sqrt{6})}{81}}$

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor $9$ out of $81$ .

$\sqrt{\frac{9 (- \sqrt{3} + \sqrt{6})}{9 (9)}}$

Step 8.4.2

Cancel the common factor.

$\sqrt{\frac{9 (- \sqrt{3} + \sqrt{6})}{9 \cdot 9}}$

Step 8.4.3

Rewrite the expression.

$\sqrt{\frac{- \sqrt{3} + \sqrt{6}}{9}}$

Step 9

Rewrite $\sqrt{\frac{- \sqrt{3} + \sqrt{6}}{9}}$ as $\frac{\sqrt{- \sqrt{3} + \sqrt{6}}}{\sqrt{9}}$ .

$\frac{\sqrt{- \sqrt{3} + \sqrt{6}}}{\sqrt{9}}$

Step 10

Simplify the denominator.

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Step 10.1

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

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

Step 10.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{- \sqrt{3} + \sqrt{6}}}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{- \sqrt{3} + \sqrt{6}}}{3}$

Decimal Form:

$0.28233922 \dots$