Simplify ( square root of 40- square root of -98)-( square root of 90+ square root of -32)

$(\sqrt{40} - \sqrt{- 98}) - (\sqrt{90} + \sqrt{- 32})$

Step 1

Simplify each term.

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$\sqrt{4 (10)} - \sqrt{- 98} - (\sqrt{90} + \sqrt{- 32})$

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

$\sqrt{2^{2} \cdot 10} - \sqrt{- 98} - (\sqrt{90} + \sqrt{- 32})$

Pull terms out from under the radical.

$2 \sqrt{10} - \sqrt{- 98} - (\sqrt{90} + \sqrt{- 32})$

Rewrite $- 98$ as $- 1 (98)$ .

$2 \sqrt{10} - \sqrt{- 1 (98)} - (\sqrt{90} + \sqrt{- 32})$

Rewrite $\sqrt{- 1 (98)}$ as $\sqrt{- 1} \cdot \sqrt{98}$ .

$2 \sqrt{10} - (\sqrt{- 1} \cdot \sqrt{98}) - (\sqrt{90} + \sqrt{- 32})$

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

$2 \sqrt{10} - (i \cdot \sqrt{98}) - (\sqrt{90} + \sqrt{- 32})$

Rewrite $98$ as $7^{2} \cdot 2$ .

Factor $49$ out of $98$ .

$2 \sqrt{10} - (i \cdot \sqrt{49 (2)}) - (\sqrt{90} + \sqrt{- 32})$

Rewrite $49$ as $7^{2}$ .

$2 \sqrt{10} - (i \cdot \sqrt{7^{2} \cdot 2}) - (\sqrt{90} + \sqrt{- 32})$

Pull terms out from under the radical.

$2 \sqrt{10} - (i \cdot (7 \sqrt{2})) - (\sqrt{90} + \sqrt{- 32})$

Move $7$ to the left of $i$ .

$2 \sqrt{10} - (7 \cdot i \sqrt{2}) - (\sqrt{90} + \sqrt{- 32})$

Multiply $7$ by $- 1$ .

$2 \sqrt{10} - 7 (i \sqrt{2}) - (\sqrt{90} + \sqrt{- 32})$

Rewrite $- 32$ as $- 1 (32)$ .

$2 \sqrt{10} - 7 i \sqrt{2} - (\sqrt{90} + \sqrt{- 1 (32)})$

Rewrite $\sqrt{- 1 (32)}$ as $\sqrt{- 1} \cdot \sqrt{32}$ .

$2 \sqrt{10} - 7 i \sqrt{2} - (\sqrt{90} + \sqrt{- 1} \cdot \sqrt{32})$

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

$2 \sqrt{10} - 7 i \sqrt{2} - (\sqrt{90} + i \cdot \sqrt{32})$

Simplify each term.

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

$2 \sqrt{10} - 7 i \sqrt{2} - (\sqrt{9 (10)} + i \cdot \sqrt{32})$

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

$2 \sqrt{10} - 7 i \sqrt{2} - (\sqrt{3^{2} \cdot 10} + i \cdot \sqrt{32})$

Pull terms out from under the radical.

$2 \sqrt{10} - 7 i \sqrt{2} - (3 \sqrt{10} + i \cdot \sqrt{32})$

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$2 \sqrt{10} - 7 i \sqrt{2} - (3 \sqrt{10} + i \cdot \sqrt{16 (2)})$

Rewrite $16$ as $4^{2}$ .

$2 \sqrt{10} - 7 i \sqrt{2} - (3 \sqrt{10} + i \cdot \sqrt{4^{2} \cdot 2})$

Pull terms out from under the radical.

$2 \sqrt{10} - 7 i \sqrt{2} - (3 \sqrt{10} + i \cdot (4 \sqrt{2}))$

Move $4$ to the left of $i$ .

$2 \sqrt{10} - 7 i \sqrt{2} - (3 \sqrt{10} + 4 i \sqrt{2})$

Apply the distributive property.

$2 \sqrt{10} - 7 i \sqrt{2} - (3 \sqrt{10}) - (4 i \sqrt{2})$

Multiply $3$ by $- 1$ .

$2 \sqrt{10} - 7 i \sqrt{2} - 3 \sqrt{10} - (4 i \sqrt{2})$

Multiply $4$ by $- 1$ .

$2 \sqrt{10} - 7 i \sqrt{2} - 3 \sqrt{10} - 4 i \sqrt{2}$

Step 2

Simplify by adding terms.

Subtract $3 \sqrt{10}$ from $2 \sqrt{10}$ .

$- 7 i \sqrt{2} - \sqrt{10} - 4 i \sqrt{2}$

Subtract $4 i \sqrt{2}$ from $- 7 i \sqrt{2}$ .

$- 11 i \sqrt{2} - \sqrt{10}$

Reorder $- 11 i \sqrt{2}$ and $- \sqrt{10}$ .

$- \sqrt{10} - 11 i \sqrt{2}$

Simplify ( square root of 40- square root of -98)-( square root of 90+ square root of -32)

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