Simplify 5 square root of -2(4+ square root of -10)

$5 \sqrt{- 2} (4 + \sqrt{- 10})$

Step 1

Rewrite $- 2$ as $- 1 (2)$ .

$5 \sqrt{- 1 (2)} (4 + \sqrt{- 10})$

Step 2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$5 (\sqrt{- 1} \cdot \sqrt{2}) (4 + \sqrt{- 10})$

Step 3

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

$5 (i \cdot \sqrt{2}) (4 + \sqrt{- 10})$

Step 4

Rewrite $- 10$ as $- 1 (10)$ .

$5 (i \cdot \sqrt{2}) (4 + \sqrt{- 1 (10)})$

Step 5

Rewrite $\sqrt{- 1 (10)}$ as $\sqrt{- 1} \cdot \sqrt{10}$ .

$5 (i \cdot \sqrt{2}) (4 + \sqrt{- 1} \cdot \sqrt{10})$

Step 6

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

$5 (i \cdot \sqrt{2}) (4 + i \cdot \sqrt{10})$

Step 7

Apply the distributive property.

$5 i \sqrt{2} \cdot 4 + 5 i \sqrt{2} (i \sqrt{10})$

Step 8

Multiply $4$ by $5$ .

$20 i \sqrt{2} + 5 i \sqrt{2} (i \sqrt{10})$

Step 9

Multiply $5 i \sqrt{2} (i \sqrt{10})$ .

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

$20 i \sqrt{2} + 5 (i^{1} i) \sqrt{2} \sqrt{10}$

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

$20 i \sqrt{2} + 5 (i^{1} i^{1}) \sqrt{2} \sqrt{10}$

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

$20 i \sqrt{2} + 5 i^{1 + 1} \sqrt{2} \sqrt{10}$

Add $1$ and $1$ .

$20 i \sqrt{2} + 5 i^{2} \sqrt{2} \sqrt{10}$

Combine using the product rule for radicals.

$20 i \sqrt{2} + 5 i^{2} \sqrt{10 \cdot 2}$

Multiply $10$ by $2$ .

$20 i \sqrt{2} + 5 i^{2} \sqrt{20}$

Step 10

Simplify each term.

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

$20 i \sqrt{2} + 5 \cdot - 1 \sqrt{20}$

Multiply $5$ by $- 1$ .

$20 i \sqrt{2} - 5 \sqrt{20}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$20 i \sqrt{2} - 5 \sqrt{4 (5)}$

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

$20 i \sqrt{2} - 5 \sqrt{2^{2} \cdot 5}$

Pull terms out from under the radical.

$20 i \sqrt{2} - 5 (2 \sqrt{5})$

Multiply $2$ by $- 5$ .

$20 i \sqrt{2} - 10 \sqrt{5}$

Step 11

Reorder $20 i \sqrt{2}$ and $- 10 \sqrt{5}$ .

$- 10 \sqrt{5} + 20 i \sqrt{2}$

Simplify 5 square root of -2(4+ square root of -10)

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