Simplify (i square root of -36)( square root of -200)

$(i \sqrt{- 36}) (\sqrt{- 200})$

Step 1

Rewrite $- 36$ as $- 1 (36)$ .

$i \sqrt{- 1 (36)} \sqrt{- 200}$

Step 2

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$i (\sqrt{- 1} \cdot \sqrt{36}) \sqrt{- 200}$

Step 3

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

$i (i \cdot \sqrt{36}) \sqrt{- 200}$

Step 4

Rewrite $- 200$ as $- 1 (200)$ .

$i (i \cdot \sqrt{36}) \sqrt{- 1 (200)}$

Step 5

Rewrite $\sqrt{- 1 (200)}$ as $\sqrt{- 1} \cdot \sqrt{200}$ .

$i (i \cdot \sqrt{36}) (\sqrt{- 1} \cdot \sqrt{200})$

Step 6

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

$i (i \cdot \sqrt{36}) (i \cdot \sqrt{200})$

Step 7

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

$i (i \cdot \sqrt{6^{2}}) (i \cdot \sqrt{200})$

Step 8

Pull terms out from under the radical, assuming positive real numbers.

$i (i \cdot 6) (i \cdot \sqrt{200})$

Step 9

Move $6$ to the left of $i$ .

$i (6 \cdot i) (i \cdot \sqrt{200})$

Step 10

Multiply $i (6 i)$ .

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

$6 (i^{1} i) (i \cdot \sqrt{200})$

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

$6 (i^{1} i^{1}) (i \cdot \sqrt{200})$

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

$6 i^{1 + 1} (i \cdot \sqrt{200})$

Add $1$ and $1$ .

$6 i^{2} (i \cdot \sqrt{200})$

Step 11

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

$6 \cdot - 1 (i \cdot \sqrt{200})$

Step 12

Multiply $6$ by $- 1$ .

$- 6 (i \cdot \sqrt{200})$

Step 13

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$- 6 (i \cdot \sqrt{100 (2)})$

Rewrite $100$ as $10^{2}$ .

$- 6 (i \cdot \sqrt{10^{2} \cdot 2})$

Step 14

Pull terms out from under the radical.

$- 6 (i \cdot (10 \sqrt{2}))$

Step 15

Simplify the expression.

Move $10$ to the left of $i$ .

$- 6 (10 \cdot i \sqrt{2})$

Multiply $10$ by $- 6$ .

$- 60 i \sqrt{2}$

Simplify (i square root of -36)( square root of -200)

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