Simplify square root of -8-2 square root of -50+ square root of 27-i square root of 108

$\sqrt{- 8} - 2 \sqrt{- 50} + \sqrt{27} - i \sqrt{108}$

Step 1

Simplify each term.

Rewrite $- 8$ as $- 1 (8)$ .

$\sqrt{- 1 (8)} - 2 \sqrt{- 50} + \sqrt{27} - i \sqrt{108}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$\sqrt{- 1} \cdot \sqrt{8} - 2 \sqrt{- 50} + \sqrt{27} - i \sqrt{108}$

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

$i \cdot \sqrt{8} - 2 \sqrt{- 50} + \sqrt{27} - i \sqrt{108}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$i \cdot \sqrt{4 (2)} - 2 \sqrt{- 50} + \sqrt{27} - i \sqrt{108}$

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

$i \cdot \sqrt{2^{2} \cdot 2} - 2 \sqrt{- 50} + \sqrt{27} - i \sqrt{108}$

Pull terms out from under the radical.

$i \cdot (2 \sqrt{2}) - 2 \sqrt{- 50} + \sqrt{27} - i \sqrt{108}$

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

$2 \cdot i \sqrt{2} - 2 \sqrt{- 50} + \sqrt{27} - i \sqrt{108}$

Rewrite $- 50$ as $- 1 (50)$ .

$2 i \sqrt{2} - 2 \sqrt{- 1 (50)} + \sqrt{27} - i \sqrt{108}$

Rewrite $\sqrt{- 1 (50)}$ as $\sqrt{- 1} \cdot \sqrt{50}$ .

$2 i \sqrt{2} - 2 (\sqrt{- 1} \cdot \sqrt{50}) + \sqrt{27} - i \sqrt{108}$

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

$2 i \sqrt{2} - 2 (i \cdot \sqrt{50}) + \sqrt{27} - i \sqrt{108}$

Rewrite $50$ as $5^{2} \cdot 2$ .

Factor $25$ out of $50$ .

$2 i \sqrt{2} - 2 (i \cdot \sqrt{25 (2)}) + \sqrt{27} - i \sqrt{108}$

Rewrite $25$ as $5^{2}$ .

$2 i \sqrt{2} - 2 (i \cdot \sqrt{5^{2} \cdot 2}) + \sqrt{27} - i \sqrt{108}$

Pull terms out from under the radical.

$2 i \sqrt{2} - 2 (i \cdot (5 \sqrt{2})) + \sqrt{27} - i \sqrt{108}$

Move $5$ to the left of $i$ .

$2 i \sqrt{2} - 2 (5 \cdot i \sqrt{2}) + \sqrt{27} - i \sqrt{108}$

Multiply $5$ by $- 2$ .

$2 i \sqrt{2} - 10 (i \sqrt{2}) + \sqrt{27} - i \sqrt{108}$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$2 i \sqrt{2} - 10 i \sqrt{2} + \sqrt{9 (3)} - i \sqrt{108}$

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

$2 i \sqrt{2} - 10 i \sqrt{2} + \sqrt{3^{2} \cdot 3} - i \sqrt{108}$

Pull terms out from under the radical.

$2 i \sqrt{2} - 10 i \sqrt{2} + 3 \sqrt{3} - i \sqrt{108}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$2 i \sqrt{2} - 10 i \sqrt{2} + 3 \sqrt{3} - i \sqrt{36 (3)}$

Rewrite $36$ as $6^{2}$ .

$2 i \sqrt{2} - 10 i \sqrt{2} + 3 \sqrt{3} - i \sqrt{6^{2} \cdot 3}$

Pull terms out from under the radical.

$2 i \sqrt{2} - 10 i \sqrt{2} + 3 \sqrt{3} - i (6 \sqrt{3})$

Multiply $6$ by $- 1$ .

$2 i \sqrt{2} - 10 i \sqrt{2} + 3 \sqrt{3} - 6 i \sqrt{3}$

Step 2

Simplify by adding terms.

Subtract $10 i \sqrt{2}$ from $2 i \sqrt{2}$ .

$- 8 i \sqrt{2} + 3 \sqrt{3} - 6 i \sqrt{3}$

Reorder $- 8 i \sqrt{2}$ and $3 \sqrt{3}$ .

$3 \sqrt{3} - 8 i \sqrt{2} - 6 i \sqrt{3}$

Simplify square root of -8-2 square root of -50+ square root of 27-i square root of 108

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