Trigonometry Examples

$\sqrt{- \frac{162}{53}}$

Step 1

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

$\sqrt{i^{2} \frac{162}{53}}$

Step 2

Pull terms out from under the radical.

$i \sqrt{\frac{162}{53}}$

Step 3

Rewrite $\sqrt{\frac{162}{53}}$ as $\frac{\sqrt{162}}{\sqrt{53}}$ .

$i \frac{\sqrt{162}}{\sqrt{53}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $162$ as $9^{2} \cdot 2$ .

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Step 4.1.1

Factor $81$ out of $162$ .

$i \frac{\sqrt{81 (2)}}{\sqrt{53}}$

Step 4.1.2

Rewrite $81$ as $9^{2}$ .

$i \frac{\sqrt{9^{2} \cdot 2}}{\sqrt{53}}$

Step 4.2

Pull terms out from under the radical.

$i \frac{9 \sqrt{2}}{\sqrt{53}}$

Step 5

Multiply $\frac{9 \sqrt{2}}{\sqrt{53}}$ by $\frac{\sqrt{53}}{\sqrt{53}}$ .

$i (\frac{9 \sqrt{2}}{\sqrt{53}} \cdot \frac{\sqrt{53}}{\sqrt{53}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{9 \sqrt{2}}{\sqrt{53}}$ by $\frac{\sqrt{53}}{\sqrt{53}}$ .

$i \frac{9 \sqrt{2} \sqrt{53}}{\sqrt{53} \sqrt{53}}$

Step 6.2

Raise $\sqrt{53}$ to the power of $1$ .

$i \frac{9 \sqrt{2} \sqrt{53}}{{\sqrt{53}}^{1} \sqrt{53}}$

Step 6.3

Raise $\sqrt{53}$ to the power of $1$ .

$i \frac{9 \sqrt{2} \sqrt{53}}{{\sqrt{53}}^{1} {\sqrt{53}}^{1}}$

Step 6.4

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

$i \frac{9 \sqrt{2} \sqrt{53}}{{\sqrt{53}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$i \frac{9 \sqrt{2} \sqrt{53}}{{\sqrt{53}}^{2}}$

Step 6.6

Rewrite ${\sqrt{53}}^{2}$ as $53$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{53}$ as $53^{\frac{1}{2}}$ .

$i \frac{9 \sqrt{2} \sqrt{53}}{{(53^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$i \frac{9 \sqrt{2} \sqrt{53}}{53^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$i \frac{9 \sqrt{2} \sqrt{53}}{53^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$i \frac{9 \sqrt{2} \sqrt{53}}{53^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$i \frac{9 \sqrt{2} \sqrt{53}}{53^{1}}$

Step 6.6.5

Evaluate the exponent.

$i \frac{9 \sqrt{2} \sqrt{53}}{53}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$i \frac{9 \sqrt{53 \cdot 2}}{53}$

Step 7.2

Multiply $53$ by $2$ .

$i \frac{9 \sqrt{106}}{53}$

Step 8

Combine fractions.

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Step 8.1

Combine $i$ and $\frac{9 \sqrt{106}}{53}$ .

$\frac{i (9 \sqrt{106})}{53}$

Step 8.2

Move $9$ to the left of $i$ .

$\frac{9 i \sqrt{106}}{53}$