Simplify ( square root of -2)(8 square root of -7)

$(\sqrt{- 2}) (8 \sqrt{- 7})$

Step 1

Rewrite $- 2$ as $- 1 (2)$ .

$\sqrt{- 1 (2)} (8 \sqrt{- 7})$

Step 2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$\sqrt{- 1} \cdot \sqrt{2} (8 \sqrt{- 7})$

Step 3

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

$i \cdot \sqrt{2} (8 \sqrt{- 7})$

Step 4

Rewrite $- 7$ as $- 1 (7)$ .

$i \cdot \sqrt{2} (8 \sqrt{- 1 (7)})$

Step 5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$i \cdot \sqrt{2} (8 (\sqrt{- 1} \cdot \sqrt{7}))$

Step 6

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

$i \cdot \sqrt{2} (8 (i \cdot \sqrt{7}))$

Step 7

Multiply $i \sqrt{2} (8 i \sqrt{7})$ .

Raise $i$ to the power of $1$ .

$\sqrt{2} (8 (i^{1} i) \sqrt{7})$

Raise $i$ to the power of $1$ .

$\sqrt{2} (8 (i^{1} i^{1}) \sqrt{7})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{2} (8 i^{1 + 1} \sqrt{7})$

Add $1$ and $1$ .

$\sqrt{2} (8 i^{2} \sqrt{7})$

Combine using the product rule for radicals.

$8 i^{2} \sqrt{2 \cdot 7}$

Multiply $2$ by $7$ .

$8 i^{2} \sqrt{14}$

Step 8

Rewrite $i^{2}$ as $- 1$ .

$8 \cdot - 1 \sqrt{14}$

Step 9

Multiply $8$ by $- 1$ .

$- 8 \sqrt{14}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 8 \sqrt{14}$

Decimal Form:

$- 29.93325909 \dots$