Simplify ( square root of -3)(9 square root of -8)

$(\sqrt{- 3}) (9 \sqrt{- 8})$

Step 1

Rewrite $- 3$ as $- 1 (3)$ .

$\sqrt{- 1 (3)} (9 \sqrt{- 8})$

Step 2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$\sqrt{- 1} \cdot \sqrt{3} (9 \sqrt{- 8})$

Step 3

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

$i \cdot \sqrt{3} (9 \sqrt{- 8})$

Step 4

Rewrite $- 8$ as $- 1 (8)$ .

$i \cdot \sqrt{3} (9 \sqrt{- 1 (8)})$

Step 5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$i \cdot \sqrt{3} (9 (\sqrt{- 1} \cdot \sqrt{8}))$

Step 6

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

$i \cdot \sqrt{3} (9 (i \cdot \sqrt{8}))$

Step 7

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$i \sqrt{3} (9 (i \cdot \sqrt{4 (2)}))$

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

$i \sqrt{3} (9 (i \cdot \sqrt{2^{2} \cdot 2}))$

Step 8

Pull terms out from under the radical.

$i \sqrt{3} (9 (i \cdot (2 \sqrt{2})))$

Step 9

Simplify the expression.

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

$i \sqrt{3} (9 (2 \cdot i \sqrt{2}))$

Multiply $2$ by $9$ .

$i \sqrt{3} (18 (i \sqrt{2}))$

Step 10

Multiply $i \sqrt{3} (18 i \sqrt{2})$ .

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

$\sqrt{3} (18 (i^{1} i) \sqrt{2})$

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

$\sqrt{3} (18 (i^{1} i^{1}) \sqrt{2})$

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

$\sqrt{3} (18 i^{1 + 1} \sqrt{2})$

Add $1$ and $1$ .

$\sqrt{3} (18 i^{2} \sqrt{2})$

Combine using the product rule for radicals.

$18 i^{2} \sqrt{3 \cdot 2}$

Multiply $3$ by $2$ .

$18 i^{2} \sqrt{6}$

Step 11

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

$18 \cdot - 1 \sqrt{6}$

Step 12

Multiply $18$ by $- 1$ .

$- 18 \sqrt{6}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- 18 \sqrt{6}$

Decimal Form:

$- 44.09081537 \dots$

Simplify ( square root of -3)(9 square root of -8)

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