Evaluate 3 square root of -24*(2 square root of -18)

$3 \sqrt{- 24} \cdot (2 \sqrt{- 18})$

Step 1

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

$3 \sqrt{- 1 (24)} \cdot (2 \sqrt{- 18})$

Step 2

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

$3 (\sqrt{- 1} \cdot \sqrt{24}) \cdot (2 \sqrt{- 18})$

Step 3

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

$3 (i \cdot \sqrt{24}) \cdot (2 \sqrt{- 18})$

Step 4

Rewrite $- 18$ as $- 1 (18)$ .

$3 (i \cdot \sqrt{24}) \cdot (2 \sqrt{- 1 (18)})$

Step 5

Rewrite $\sqrt{- 1 (18)}$ as $\sqrt{- 1} \cdot \sqrt{18}$ .

$3 (i \cdot \sqrt{24}) \cdot (2 (\sqrt{- 1} \cdot \sqrt{18}))$

Step 6

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

$3 (i \cdot \sqrt{24}) \cdot (2 (i \cdot \sqrt{18}))$

Step 7

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

Factor $4$ out of $24$ .

$3 (i \cdot \sqrt{4 (6)}) \cdot (2 (i \cdot \sqrt{18}))$

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

$3 (i \cdot \sqrt{2^{2} \cdot 6}) \cdot (2 (i \cdot \sqrt{18}))$

Step 8

Pull terms out from under the radical.

$3 (i \cdot (2 \sqrt{6})) \cdot (2 (i \cdot \sqrt{18}))$

Step 9

Simplify the expression.

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

$3 (2 \cdot i \sqrt{6}) \cdot (2 (i \cdot \sqrt{18}))$

Multiply $2$ by $3$ .

$6 (i \sqrt{6}) \cdot (2 (i \cdot \sqrt{18}))$

Step 10

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$6 i \sqrt{6} \cdot (2 (i \cdot \sqrt{9 (2)}))$

Rewrite $9$ as $3^{2}$ .

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

Step 11

Pull terms out from under the radical.

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

Step 12

Simplify the expression.

Move $3$ to the left of $i$ .

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

Multiply $3$ by $2$ .

$6 i \sqrt{6} \cdot (6 (i \sqrt{2}))$

Step 13

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

Multiply $6$ by $6$ .

$36 i \sqrt{6} (i \sqrt{2})$

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

$36 (i^{1} i) \sqrt{6} \sqrt{2}$

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

$36 (i^{1} i^{1}) \sqrt{6} \sqrt{2}$

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

$36 i^{1 + 1} \sqrt{6} \sqrt{2}$

Add $1$ and $1$ .

$36 i^{2} \sqrt{6} \sqrt{2}$

Combine using the product rule for radicals.

$36 i^{2} \sqrt{2 \cdot 6}$

Multiply $2$ by $6$ .

$36 i^{2} \sqrt{12}$

Step 14

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

$36 \cdot - 1 \sqrt{12}$

Step 15

Multiply $36$ by $- 1$ .

$- 36 \sqrt{12}$

Step 16

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$- 36 \sqrt{4 (3)}$

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

$- 36 \sqrt{2^{2} \cdot 3}$

Step 17

Pull terms out from under the radical.

$- 36 (2 \sqrt{3})$

Step 18

Multiply $2$ by $- 36$ .

$- 72 \sqrt{3}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$- 72 \sqrt{3}$

Decimal Form:

$- 124.70765814 \dots$

Evaluate 3 square root of -24*(2 square root of -18)

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