Trigonometry Examples

$\sqrt{\frac{1 - (- \frac{\sqrt{32}}{9})}{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 1.1.1

Factor $16$ out of $32$ .

$\sqrt{\frac{1 - - \frac{\sqrt{16 (2)}}{9}}{2}}$

Step 1.1.2

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

$\sqrt{\frac{1 - - \frac{\sqrt{4^{2} \cdot 2}}{9}}{2}}$

Step 1.2

Pull terms out from under the radical.

$\sqrt{\frac{1 - - \frac{4 \sqrt{2}}{9}}{2}}$

Step 2

Multiply $- - \frac{4 \sqrt{2}}{9}$ .

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Step 2.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{1 + 1 \frac{4 \sqrt{2}}{9}}{2}}$

Step 2.2

Multiply $\frac{4 \sqrt{2}}{9}$ by $1$ .

$\sqrt{\frac{1 + \frac{4 \sqrt{2}}{9}}{2}}$

Step 3

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{9}{9} + \frac{4 \sqrt{2}}{9}}{2}}$

Step 4

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{9 + 4 \sqrt{2}}{9}}{2}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{9 + 4 \sqrt{2}}{9} \cdot \frac{1}{2}}$

Step 6

Multiply $\frac{9 + 4 \sqrt{2}}{9} \cdot \frac{1}{2}$ .

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Step 6.1

Multiply $\frac{9 + 4 \sqrt{2}}{9}$ by $\frac{1}{2}$ .

$\sqrt{\frac{9 + 4 \sqrt{2}}{9 \cdot 2}}$

Step 6.2

Multiply $9$ by $2$ .

$\sqrt{\frac{9 + 4 \sqrt{2}}{18}}$

Step 7

Rewrite $\sqrt{\frac{9 + 4 \sqrt{2}}{18}}$ as $\frac{\sqrt{9 + 4 \sqrt{2}}}{\sqrt{18}}$ .

$\frac{\sqrt{9 + 4 \sqrt{2}}}{\sqrt{18}}$

Step 8

Simplify the denominator.

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Step 8.1

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

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Step 8.1.1

Factor $9$ out of $18$ .

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

Step 8.1.2

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

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

Step 8.2

Pull terms out from under the radical.

$\frac{\sqrt{9 + 4 \sqrt{2}}}{3 \sqrt{2}}$

Step 9

Multiply $\frac{\sqrt{9 + 4 \sqrt{2}}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{\sqrt{9 + 4 \sqrt{2}}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{9 + 4 \sqrt{2}} \sqrt{2}}{3 \sqrt{2} \sqrt{2}}$

Step 10.2

Move $\sqrt{2}$ .

$\frac{\sqrt{9 + 4 \sqrt{2}} \sqrt{2}}{3 (\sqrt{2} \sqrt{2})}$

Step 10.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{9 + 4 \sqrt{2}} \sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 10.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{9 + 4 \sqrt{2}} \sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 10.5

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

$\frac{\sqrt{9 + 4 \sqrt{2}} \sqrt{2}}{3 {\sqrt{2}}^{1 + 1}}$

Step 10.6

Add $1$ and $1$ .

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

Step 10.7

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

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Step 10.7.1

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

$\frac{\sqrt{9 + 4 \sqrt{2}} \sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}}$

Step 10.7.2

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

$\frac{\sqrt{9 + 4 \sqrt{2}} \sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 10.7.3

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

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

Step 10.7.4

Cancel the common factor of $2$ .

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Step 10.7.4.1

Cancel the common factor.

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

Step 10.7.4.2

Rewrite the expression.

$\frac{\sqrt{9 + 4 \sqrt{2}} \sqrt{2}}{3 \cdot 2^{1}}$

Step 10.7.5

Evaluate the exponent.

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

Step 11

Combine using the product rule for radicals.

$\frac{\sqrt{(9 + 4 \sqrt{2}) \cdot 2}}{3 \cdot 2}$

Step 12

Multiply $3$ by $2$ .

$\frac{\sqrt{(9 + 4 \sqrt{2}) \cdot 2}}{6}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(9 + 4 \sqrt{2}) \cdot 2}}{6}$

Decimal Form:

$0.90236892 \dots$