Simplify the Radical Expression (-8 square root of 30)(3 square root of -24)

$(- 8 \sqrt{30}) (3 \sqrt{- 24})$

Rewrite $- 24$ as $- 1 (24)$ .

$- 8 \sqrt{30} (3 \sqrt{- 1 (24)})$

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$- 8 \sqrt{30} (3 (\sqrt{- 1} \cdot \sqrt{24}))$

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

$- 8 \sqrt{30} (3 (i \cdot \sqrt{24}))$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$- 8 \sqrt{30} (3 (i \cdot \sqrt{4 (6)}))$

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

$- 8 \sqrt{30} (3 (i \cdot \sqrt{2^{2} \cdot 6}))$

Pull terms out from under the radical.

$- 8 \sqrt{30} (3 (i \cdot (| 2 | \sqrt{6})))$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$- 8 \sqrt{30} (3 (i \cdot (2 \sqrt{6})))$

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

$- 8 \sqrt{30} (3 (2 \cdot i \sqrt{6}))$

Multiply $2$ by $3$ .

$- 8 \sqrt{30} (6 (i \sqrt{6}))$

Multiply $- 8 \sqrt{30} (6 i \sqrt{6})$ .

Multiply $6$ by $- 8$ .

$- 48 \sqrt{30} (i \sqrt{6})$

Combine using the product rule for radicals.

$- 48 \sqrt{6 \cdot 30} i$

Multiply $6$ by $30$ .

$- 48 \sqrt{180} i$

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$- 48 \sqrt{36 (5)} i$

Rewrite $36$ as $6^{2}$ .

$- 48 \sqrt{6^{2} \cdot 5} i$

Pull terms out from under the radical.

$- 48 (| 6 | \sqrt{5}) i$

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$- 48 (6 \sqrt{5}) i$

Multiply $6$ by $- 48$ .

$- 288 \sqrt{5} i$