Simplify (- square root of -6)/( square root of 18+ square root of -4)

$\frac{- \sqrt{- 6}}{\sqrt{18} + \sqrt{- 4}}$

Step 1

Pull out imaginary unit $i$ .

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- \sqrt{- 1 (6)}}{\sqrt{18} + \sqrt{- 4}}$

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$\frac{- (\sqrt{- 1} \cdot \sqrt{6})}{\sqrt{18} + \sqrt{- 4}}$

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

$\frac{- (i \cdot \sqrt{6})}{\sqrt{18} + \sqrt{- 4}}$

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- (i \cdot \sqrt{6})}{\sqrt{18} + \sqrt{- 1 (4)}}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$\frac{- (i \cdot \sqrt{6})}{\sqrt{18} + \sqrt{- 1} \cdot \sqrt{4}}$

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

$\frac{- (i \cdot \sqrt{6})}{\sqrt{18} + i \cdot \sqrt{4}}$

Step 2

Simplify the denominator.

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

Factor $9$ out of $18$ .

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

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

$\frac{- i \cdot \sqrt{6}}{\sqrt{3^{2} \cdot 2} + i \cdot \sqrt{4}}$

Pull terms out from under the radical.

$\frac{- i \cdot \sqrt{6}}{3 \sqrt{2} + i \cdot \sqrt{4}}$

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

$\frac{- i \cdot \sqrt{6}}{3 \sqrt{2} + i \cdot \sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{- i \cdot \sqrt{6}}{3 \sqrt{2} + i \cdot 2}$

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

$\frac{- i \cdot \sqrt{6}}{3 \sqrt{2} + 2 i}$

Step 3

Move the negative in front of the fraction.

$- \frac{i \cdot \sqrt{6}}{3 \sqrt{2} + 2 i}$

Step 4

Multiply $\frac{i \cdot \sqrt{6}}{3 \sqrt{2} + 2 i}$ by $\frac{3 \sqrt{2} - 2 i}{3 \sqrt{2} - 2 i}$ .

$- (\frac{i \cdot \sqrt{6}}{3 \sqrt{2} + 2 i} \cdot \frac{3 \sqrt{2} - 2 i}{3 \sqrt{2} - 2 i})$

Step 5

Multiply $\frac{i \cdot \sqrt{6}}{3 \sqrt{2} + 2 i}$ by $\frac{3 \sqrt{2} - 2 i}{3 \sqrt{2} - 2 i}$ .

$- \frac{i \cdot \sqrt{6} (3 \sqrt{2} - 2 i)}{(3 \sqrt{2} + 2 i) (3 \sqrt{2} - 2 i)}$

Step 6

Expand the denominator using the FOIL method.

$- \frac{i \cdot \sqrt{6} (3 \sqrt{2} - 2 i)}{9 {\sqrt{2}}^{2} - 6 \sqrt{2} i + 6 i \sqrt{2} - 4 i^{2}}$

Step 7

Simplify.

$- \frac{i \cdot \sqrt{6} (3 \sqrt{2} - 2 i)}{22}$

Step 8

Group $3 \sqrt{2} - 2 i$ and $i$ together.

$- \frac{(3 \sqrt{2} - 2 i) i \cdot \sqrt{6}}{22}$

Step 9

Apply the distributive property.

$- \frac{(3 \sqrt{2} i - 2 i i) \cdot \sqrt{6}}{22}$

Step 10

Multiply $- 2 i i$ .

Raise $i$ to the power of $1$ .

$- \frac{(3 \sqrt{2} i - 2 (i^{1} i)) \cdot \sqrt{6}}{22}$

Raise $i$ to the power of $1$ .

$- \frac{(3 \sqrt{2} i - 2 (i^{1} i^{1})) \cdot \sqrt{6}}{22}$

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

$- \frac{(3 \sqrt{2} i - 2 i^{1 + 1}) \cdot \sqrt{6}}{22}$

Add $1$ and $1$ .

$- \frac{(3 \sqrt{2} i - 2 i^{2}) \cdot \sqrt{6}}{22}$

Step 11

Simplify each term.

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

$- \frac{(3 \sqrt{2} i - 2 \cdot - 1) \cdot \sqrt{6}}{22}$

Multiply $- 2$ by $- 1$ .

$- \frac{(3 \sqrt{2} i + 2) \cdot \sqrt{6}}{22}$

Step 12

Reorder $3 \sqrt{2} i$ and $2$ .

$- \frac{(2 + 3 \sqrt{2} i) \sqrt{6}}{22}$

Simplify (- square root of -6)/( square root of 18+ square root of -4)

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