Trigonometry Examples

$\sqrt{\frac{- 43}{6}}$

Step 1

Simplify the expression.

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Step 1.1

Move the negative in front of the fraction.

$\sqrt{- \frac{43}{6}}$

Step 1.2

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

$\sqrt{i^{2} \frac{43}{6}}$

Step 2

Pull terms out from under the radical.

$i \sqrt{\frac{43}{6}}$

Step 3

Rewrite $\sqrt{\frac{43}{6}}$ as $\frac{\sqrt{43}}{\sqrt{6}}$ .

$i \frac{\sqrt{43}}{\sqrt{6}}$

Step 4

Multiply $\frac{\sqrt{43}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$i (\frac{\sqrt{43}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt{43}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$i \frac{\sqrt{43} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 5.2

Raise $\sqrt{6}$ to the power of $1$ .

$i \frac{\sqrt{43} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 5.3

Raise $\sqrt{6}$ to the power of $1$ .

$i \frac{\sqrt{43} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 5.4

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

$i \frac{\sqrt{43} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 5.5

Add $1$ and $1$ .

$i \frac{\sqrt{43} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 5.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$i \frac{\sqrt{43} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 5.6.2

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

$i \frac{\sqrt{43} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 5.6.3

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

$i \frac{\sqrt{43} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$i \frac{\sqrt{43} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 5.6.4.2

Rewrite the expression.

$i \frac{\sqrt{43} \sqrt{6}}{6^{1}}$

Step 5.6.5

Evaluate the exponent.

$i \frac{\sqrt{43} \sqrt{6}}{6}$

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

$i \frac{\sqrt{43 \cdot 6}}{6}$

Step 6.2

Multiply $43$ by $6$ .

$i \frac{\sqrt{258}}{6}$

Step 7

Combine $i$ and $\frac{\sqrt{258}}{6}$ .

$\frac{i \sqrt{258}}{6}$