Simplify the Radical Expression (4 square root of 20)(-6 square root of -6)

$(4 \sqrt{20}) (- 6 \sqrt{- 6})$

Rewrite $- 6$ as $- 1 (6)$ .

$4 \sqrt{20} (- 6 \sqrt{- 1 (6)})$

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$4 \sqrt{20} (- 6 (\sqrt{- 1} \cdot \sqrt{6}))$

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

$4 \sqrt{20} (- 6 (i \cdot \sqrt{6}))$

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

Factor $4$ out of $20$ .

$4 \sqrt{4 (5)} (- 6 (i \cdot \sqrt{6}))$

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

$4 \sqrt{2^{2} \cdot 5} (- 6 (i \cdot \sqrt{6}))$

Pull terms out from under the radical.

$4 (| 2 | \sqrt{5}) (- 6 (i \cdot \sqrt{6}))$

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

$4 (2 \sqrt{5}) (- 6 (i \cdot \sqrt{6}))$

Multiply $2$ by $4$ .

$8 \sqrt{5} (- 6 (i \cdot \sqrt{6}))$

Multiply $8 \sqrt{5} (- 6 i \sqrt{6})$ .

Multiply $- 6$ by $8$ .

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

Combine using the product rule for radicals.

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

Multiply $6$ by $5$ .

$- 48 \sqrt{30} i$