Simplify the Radical Expression (-3 square root of 3)( square root of -30)

$(- 3 \sqrt{3}) (\sqrt{- 30})$

Rewrite $- 30$ as $- 1 (30)$ .

$- 3 \sqrt{3} \sqrt{- 1 (30)}$

Rewrite $\sqrt{- 1 (30)}$ as $\sqrt{- 1} \cdot \sqrt{30}$ .

$- 3 \sqrt{3} (\sqrt{- 1} \cdot \sqrt{30})$

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

$- 3 \sqrt{3} (i \cdot \sqrt{30})$

Multiply $- 3 \sqrt{3} (i \sqrt{30})$ .

Combine using the product rule for radicals.

$- 3 \sqrt{30 \cdot 3} i$

Multiply $30$ by $3$ .

$- 3 \sqrt{90} i$

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

Factor $9$ out of $90$ .

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

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

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

Pull terms out from under the radical.

$- 3 (| 3 | \sqrt{10}) i$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$- 3 (3 \sqrt{10}) i$

Multiply $3$ by $- 3$ .

$- 9 \sqrt{10} i$