Trigonometry Examples

$\sqrt{\frac{- 56}{65}}$

Step 1

Simplify the expression.

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Step 1.1

Move the negative in front of the fraction.

$\sqrt{- \frac{56}{65}}$

Step 1.2

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

$\sqrt{i^{2} \frac{56}{65}}$

Step 2

Pull terms out from under the radical.

$i \sqrt{\frac{56}{65}}$

Step 3

Rewrite $\sqrt{\frac{56}{65}}$ as $\frac{\sqrt{56}}{\sqrt{65}}$ .

$i \frac{\sqrt{56}}{\sqrt{65}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 4.1.1

Factor $4$ out of $56$ .

$i \frac{\sqrt{4 (14)}}{\sqrt{65}}$

Step 4.1.2

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

$i \frac{\sqrt{2^{2} \cdot 14}}{\sqrt{65}}$

Step 4.2

Pull terms out from under the radical.

$i \frac{2 \sqrt{14}}{\sqrt{65}}$

Step 5

Multiply $\frac{2 \sqrt{14}}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$i (\frac{2 \sqrt{14}}{\sqrt{65}} \cdot \frac{\sqrt{65}}{\sqrt{65}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{2 \sqrt{14}}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$i \frac{2 \sqrt{14} \sqrt{65}}{\sqrt{65} \sqrt{65}}$

Step 6.2

Raise $\sqrt{65}$ to the power of $1$ .

$i \frac{2 \sqrt{14} \sqrt{65}}{{\sqrt{65}}^{1} \sqrt{65}}$

Step 6.3

Raise $\sqrt{65}$ to the power of $1$ .

$i \frac{2 \sqrt{14} \sqrt{65}}{{\sqrt{65}}^{1} {\sqrt{65}}^{1}}$

Step 6.4

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

$i \frac{2 \sqrt{14} \sqrt{65}}{{\sqrt{65}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$i \frac{2 \sqrt{14} \sqrt{65}}{{\sqrt{65}}^{2}}$

Step 6.6

Rewrite ${\sqrt{65}}^{2}$ as $65$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{65}$ as $65^{\frac{1}{2}}$ .

$i \frac{2 \sqrt{14} \sqrt{65}}{{(65^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$i \frac{2 \sqrt{14} \sqrt{65}}{65^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$i \frac{2 \sqrt{14} \sqrt{65}}{65^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$i \frac{2 \sqrt{14} \sqrt{65}}{65^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$i \frac{2 \sqrt{14} \sqrt{65}}{65^{1}}$

Step 6.6.5

Evaluate the exponent.

$i \frac{2 \sqrt{14} \sqrt{65}}{65}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$i \frac{2 \sqrt{65 \cdot 14}}{65}$

Step 7.2

Multiply $65$ by $14$ .

$i \frac{2 \sqrt{910}}{65}$

Step 8

Combine fractions.

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Step 8.1

Combine $i$ and $\frac{2 \sqrt{910}}{65}$ .

$\frac{i (2 \sqrt{910})}{65}$

Step 8.2

Move $2$ to the left of $i$ .

$\frac{2 i \sqrt{910}}{65}$