Simplify 5 square root of -100+4 cube root of 512+2 square root of -64

$5 \sqrt{- 100} + 4 \sqrt[3]{512} + 2 \sqrt{- 64}$

Simplify each term.

Rewrite $- 100$ as $- 1 (100)$ .

$5 \sqrt{- 1 (100)} + 4 \sqrt[3]{512} + 2 \sqrt{- 64}$

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$5 (\sqrt{- 1} \cdot \sqrt{100}) + 4 \sqrt[3]{512} + 2 \sqrt{- 64}$

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

$5 (i \cdot \sqrt{100}) + 4 \sqrt[3]{512} + 2 \sqrt{- 64}$

Rewrite $100$ as $10^{2}$ .

$5 (i \cdot \sqrt{10^{2}}) + 4 \sqrt[3]{512} + 2 \sqrt{- 64}$

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

$5 (i \cdot 10) + 4 \sqrt[3]{512} + 2 \sqrt{- 64}$

Move $10$ to the left of $i$ .

$5 (10 \cdot i) + 4 \sqrt[3]{512} + 2 \sqrt{- 64}$

Multiply $10$ by $5$ .

$50 i + 4 \sqrt[3]{512} + 2 \sqrt{- 64}$

Rewrite $512$ as $8^{3}$ .

$50 i + 4 \sqrt[3]{8^{3}} + 2 \sqrt{- 64}$

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

$50 i + 4 \cdot 8 + 2 \sqrt{- 64}$

Multiply $4$ by $8$ .

$50 i + 32 + 2 \sqrt{- 64}$

Rewrite $- 64$ as $- 1 (64)$ .

$50 i + 32 + 2 \sqrt{- 1 (64)}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$50 i + 32 + 2 (\sqrt{- 1} \cdot \sqrt{64})$

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

$50 i + 32 + 2 (i \cdot \sqrt{64})$

Rewrite $64$ as $8^{2}$ .

$50 i + 32 + 2 (i \cdot \sqrt{8^{2}})$

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

$50 i + 32 + 2 (i \cdot 8)$

Move $8$ to the left of $i$ .

$50 i + 32 + 2 (8 \cdot i)$

Multiply $8$ by $2$ .

$50 i + 32 + 16 i$

Add $50 i$ and $16 i$ .

$32 + 66 i$