Simplify (3 square root of 6)/( square root of -12)

$\frac{3 \sqrt{6}}{\sqrt{- 12}}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 12$ as $- 1 (12)$ .

$\frac{3 \sqrt{6}}{\sqrt{- 1 (12)}}$

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$\frac{3 \sqrt{6}}{\sqrt{- 1} \cdot \sqrt{12}}$

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{3 \sqrt{6}}{i \cdot \sqrt{12}}$

Step 2

Combine $\sqrt{6}$ and $\sqrt{12}$ into a single radical.

$\frac{3 \sqrt{\frac{6}{12}}}{i}$

Step 3

Cancel the common factor of $6$ and $12$ .

Factor $6$ out of $6$ .

$\frac{3 \sqrt{\frac{6 (1)}{12}}}{i}$

Cancel the common factors.

Factor $6$ out of $12$ .

$\frac{3 \sqrt{\frac{6 \cdot 1}{6 \cdot 2}}}{i}$

Cancel the common factor.

$\frac{3 \sqrt{\frac{6 \cdot 1}{6 \cdot 2}}}{i}$

Rewrite the expression.

$\frac{3 \sqrt{\frac{1}{2}}}{i}$

Step 4

Simplify the numerator.

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{3 \frac{\sqrt{1}}{\sqrt{2}}}{i}$

Any root of $1$ is $1$ .

$\frac{3 \frac{1}{\sqrt{2}}}{i}$

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

$\frac{3 (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}{i}$

Combine and simplify the denominator.

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

$\frac{3 \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}}{i}$

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

$\frac{3 \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}{i}$

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

$\frac{3 \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}{i}$

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

$\frac{3 \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}}{i}$

Add $1$ and $1$ .

$\frac{3 \frac{\sqrt{2}}{{\sqrt{2}}^{2}}}{i}$

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

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

$\frac{3 \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}{i}$

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

$\frac{3 \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}{i}$

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

$\frac{3 \frac{\sqrt{2}}{2^{\frac{2}{2}}}}{i}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 \frac{\sqrt{2}}{2^{\frac{2}{2}}}}{i}$

Rewrite the expression.

$\frac{3 \frac{\sqrt{2}}{2^{1}}}{i}$

Evaluate the exponent.

$\frac{3 \frac{\sqrt{2}}{2}}{i}$

Step 5

Combine $3$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{3 \sqrt{2}}{2}}{i}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 \sqrt{2}}{2} \cdot \frac{1}{i}$

Step 7

Multiply $\frac{3 \sqrt{2}}{2}$ by $\frac{1}{i}$ .

$\frac{3 \sqrt{2}}{2 i}$

Simplify (3 square root of 6)/( square root of -12)

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