Algebra Examples

$(i \sqrt{- 32}) (i \sqrt{- 27})$

Step 1

Rewrite $- 32$ as $- 1 (32)$ .

$i \sqrt{- 1 (32)} (i \sqrt{- 27})$

Step 2

Rewrite $\sqrt{- 1 (32)}$ as $\sqrt{- 1} \cdot \sqrt{32}$ .

$i (\sqrt{- 1} \cdot \sqrt{32}) (i \sqrt{- 27})$

Step 3

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

$i (i \cdot \sqrt{32}) (i \sqrt{- 27})$

Step 4

Rewrite $- 27$ as $- 1 (27)$ .

$i (i \cdot \sqrt{32}) (i \sqrt{- 1 (27)})$

Step 5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$i (i \cdot \sqrt{32}) (i (\sqrt{- 1} \cdot \sqrt{27}))$

Step 6

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

$i (i \cdot \sqrt{32}) (i (i \cdot \sqrt{27}))$

Step 7

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 7.1

Factor $16$ out of $32$ .

$i (i \cdot \sqrt{16 (2)}) (i (i \cdot \sqrt{27}))$

Step 7.2

Rewrite $16$ as $4^{2}$ .

$i (i \cdot \sqrt{4^{2} \cdot 2}) (i (i \cdot \sqrt{27}))$

Step 8

Pull terms out from under the radical.

$i (i \cdot (4 \sqrt{2})) (i (i \cdot \sqrt{27}))$

Step 9

Move $4$ to the left of $i$ .

$i (4 \cdot i \sqrt{2}) (i (i \cdot \sqrt{27}))$

Step 10

Multiply $i (4 i \sqrt{2})$ .

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Step 10.1

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

$4 (i^{1} i) \sqrt{2} (i (i \cdot \sqrt{27}))$

Step 10.2

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

$4 (i^{1} i^{1}) \sqrt{2} (i (i \cdot \sqrt{27}))$

Step 10.3

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

$4 i^{1 + 1} \sqrt{2} (i (i \cdot \sqrt{27}))$

Step 10.4

Add $1$ and $1$ .

$4 i^{2} \sqrt{2} (i (i \cdot \sqrt{27}))$

Step 11

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

$4 \cdot - 1 \sqrt{2} (i (i \cdot \sqrt{27}))$

Step 12

Multiply $4$ by $- 1$ .

$- 4 \sqrt{2} (i (i \cdot \sqrt{27}))$

Step 13

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 13.1

Factor $9$ out of $27$ .

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

Step 13.2

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

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

Step 14

Pull terms out from under the radical.

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

Step 15

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

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

Step 16

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

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Step 16.1

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

$- 4 \sqrt{2} (3 (i^{1} i) \sqrt{3})$

Step 16.2

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

$- 4 \sqrt{2} (3 (i^{1} i^{1}) \sqrt{3})$

Step 16.3

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

$- 4 \sqrt{2} (3 i^{1 + 1} \sqrt{3})$

Step 16.4

Add $1$ and $1$ .

$- 4 \sqrt{2} (3 i^{2} \sqrt{3})$

Step 17

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

$- 4 \sqrt{2} (3 \cdot - 1 \sqrt{3})$

Step 18

Multiply $3$ by $- 1$ .

$- 4 \sqrt{2} (- 3 \sqrt{3})$

Step 19

Multiply $- 4 \sqrt{2} (- 3 \sqrt{3})$ .

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Step 19.1

Multiply $- 3$ by $- 4$ .

$12 \sqrt{2} \sqrt{3}$

Step 19.2

Combine using the product rule for radicals.

$12 \sqrt{3 \cdot 2}$

Step 19.3

Multiply $3$ by $2$ .

$12 \sqrt{6}$

Step 20

The result can be shown in multiple forms.

Exact Form:

$12 \sqrt{6}$

Decimal Form:

$29.39387691 \dots$