Simplify the Radical Expression (-8 square root of 6)(2 square root of -42)

$(- 8 \sqrt{6}) (2 \sqrt{- 42})$

Step 1

Rewrite $- 42$ as $- 1 (42)$ .

$- 8 \sqrt{6} (2 \sqrt{- 1 (42)})$

Step 2

Rewrite $\sqrt{- 1 (42)}$ as $\sqrt{- 1} \cdot \sqrt{42}$ .

$- 8 \sqrt{6} (2 (\sqrt{- 1} \cdot \sqrt{42}))$

Step 3

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

$- 8 \sqrt{6} (2 (i \cdot \sqrt{42}))$

Step 4

Multiply $- 8 \sqrt{6} (2 i \sqrt{42})$ .

Multiply $2$ by $- 8$ .

$- 16 \sqrt{6} (i \sqrt{42})$

Combine using the product rule for radicals.

$- 16 \sqrt{42 \cdot 6} i$

Multiply $42$ by $6$ .

$- 16 \sqrt{252} i$

Step 5

Rewrite $252$ as $6^{2} \cdot 7$ .

Factor $36$ out of $252$ .

$- 16 \sqrt{36 (7)} i$

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

$- 16 \sqrt{6^{2} \cdot 7} i$

Step 6

Pull terms out from under the radical.

$- 16 (| 6 | \sqrt{7}) i$

Step 7

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

$- 16 (6 \sqrt{7}) i$

Step 8

Multiply $6$ by $- 16$ .

$- 96 \sqrt{7} i$