Simplify (i square root of -363)( square root of -243)

$(i \sqrt{- 363}) (\sqrt{- 243})$

Step 1

Rewrite $- 363$ as $- 1 (363)$ .

$i \sqrt{- 1 (363)} \sqrt{- 243}$

Step 2

Rewrite $\sqrt{- 1 (363)}$ as $\sqrt{- 1} \cdot \sqrt{363}$ .

$i (\sqrt{- 1} \cdot \sqrt{363}) \sqrt{- 243}$

Step 3

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

$i (i \cdot \sqrt{363}) \sqrt{- 243}$

Step 4

Rewrite $- 243$ as $- 1 (243)$ .

$i (i \cdot \sqrt{363}) \sqrt{- 1 (243)}$

Step 5

Rewrite $\sqrt{- 1 (243)}$ as $\sqrt{- 1} \cdot \sqrt{243}$ .

$i (i \cdot \sqrt{363}) (\sqrt{- 1} \cdot \sqrt{243})$

Step 6

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

$i (i \cdot \sqrt{363}) (i \cdot \sqrt{243})$

Step 7

Rewrite $363$ as $11^{2} \cdot 3$ .

Factor $121$ out of $363$ .

$i (i \cdot \sqrt{121 (3)}) (i \cdot \sqrt{243})$

Rewrite $121$ as $11^{2}$ .

$i (i \cdot \sqrt{11^{2} \cdot 3}) (i \cdot \sqrt{243})$

Step 8

Pull terms out from under the radical.

$i (i \cdot (11 \sqrt{3})) (i \cdot \sqrt{243})$

Step 9

Move $11$ to the left of $i$ .

$i (11 \cdot i \sqrt{3}) (i \cdot \sqrt{243})$

Step 10

Multiply $i (11 i \sqrt{3})$ .

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

$11 (i^{1} i) \sqrt{3} (i \cdot \sqrt{243})$

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

$11 (i^{1} i^{1}) \sqrt{3} (i \cdot \sqrt{243})$

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

$11 i^{1 + 1} \sqrt{3} (i \cdot \sqrt{243})$

Add $1$ and $1$ .

$11 i^{2} \sqrt{3} (i \cdot \sqrt{243})$

Step 11

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

$11 \cdot - 1 \sqrt{3} (i \cdot \sqrt{243})$

Step 12

Multiply $11$ by $- 1$ .

$- 11 \sqrt{3} (i \cdot \sqrt{243})$

Step 13

Rewrite $243$ as $9^{2} \cdot 3$ .

Factor $81$ out of $243$ .

$- 11 \sqrt{3} (i \cdot \sqrt{81 (3)})$

Rewrite $81$ as $9^{2}$ .

$- 11 \sqrt{3} (i \cdot \sqrt{9^{2} \cdot 3})$

Step 14

Pull terms out from under the radical.

$- 11 \sqrt{3} (i \cdot (9 \sqrt{3}))$

Step 15

Move $9$ to the left of $i$ .

$- 11 \sqrt{3} (9 \cdot i \sqrt{3})$

Step 16

Multiply $- 11 \sqrt{3} (9 i \sqrt{3})$ .

Multiply $9$ by $- 11$ .

$- 99 \sqrt{3} (i \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$- 99 ({\sqrt{3}}^{1} \sqrt{3}) i$

Raise $\sqrt{3}$ to the power of $1$ .

$- 99 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) i$

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

$- 99 {\sqrt{3}}^{1 + 1} i$

Add $1$ and $1$ .

$- 99 {\sqrt{3}}^{2} i$

Step 17

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- 99 {(3^{\frac{1}{2}})}^{2} i$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 99 \cdot 3^{\frac{1}{2} \cdot 2} i$

Combine $\frac{1}{2}$ and $2$ .

$- 99 \cdot 3^{\frac{2}{2}} i$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 99 \cdot 3^{\frac{2}{2}} i$

Rewrite the expression.

$- 99 \cdot 3^{1} i$

Evaluate the exponent.

$- 99 \cdot 3 i$

Step 18

Multiply $- 99$ by $3$ .

$- 297 i$

Simplify (i square root of -363)( square root of -243)

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