Simplify - cube root of 216-2 square root of -4+ cube root of 1

$- \sqrt[3]{216} - 2 \sqrt{- 4} + \sqrt[3]{1}$

Step 1

Simplify each term.

Rewrite $216$ as $6^{3}$ .

$- \sqrt[3]{6^{3}} - 2 \sqrt{- 4} + \sqrt[3]{1}$

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

$- 1 \cdot 6 - 2 \sqrt{- 4} + \sqrt[3]{1}$

Multiply $- 1$ by $6$ .

$- 6 - 2 \sqrt{- 4} + \sqrt[3]{1}$

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

$- 6 - 2 \sqrt{- 1 (4)} + \sqrt[3]{1}$

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

$- 6 - 2 (\sqrt{- 1} \cdot \sqrt{4}) + \sqrt[3]{1}$

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

$- 6 - 2 (i \cdot \sqrt{4}) + \sqrt[3]{1}$

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

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

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

$- 6 - 2 (i \cdot 2) + \sqrt[3]{1}$

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

$- 6 - 2 (2 \cdot i) + \sqrt[3]{1}$

Multiply $2$ by $- 2$ .

$- 6 - 4 i + \sqrt[3]{1}$

Any root of $1$ is $1$ .

$- 6 - 4 i + 1$

Step 2

Add $- 6$ and $1$ .

$- 5 - 4 i$