Algebra Examples

$\sqrt{- 9} \cdot \sqrt{- 16} \cdot \sqrt{32}$

Step 1

Rewrite $- 9$ as $- 1 (9)$ .

$\sqrt{- 1 (9)} \cdot \sqrt{- 16} \cdot \sqrt{32}$

Step 2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$\sqrt{- 1} \cdot \sqrt{9} \cdot \sqrt{- 16} \cdot \sqrt{32}$

Step 3

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

$i \cdot \sqrt{9} \cdot \sqrt{- 16} \cdot \sqrt{32}$

Step 4

Rewrite $- 16$ as $- 1 (16)$ .

$i \cdot \sqrt{9} \cdot \sqrt{- 1 (16)} \cdot \sqrt{32}$

Step 5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$i \cdot \sqrt{9} \cdot (\sqrt{- 1} \cdot \sqrt{16}) \cdot \sqrt{32}$

Step 6

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

$i \cdot \sqrt{9} \cdot (i \cdot \sqrt{16}) \cdot \sqrt{32}$

Step 7

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

$i \cdot \sqrt{3^{2}} \cdot (i \cdot \sqrt{16}) \cdot \sqrt{32}$

Step 8

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

$i \cdot 3 \cdot (i \cdot \sqrt{16}) \cdot \sqrt{32}$

Step 9

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

$3 \cdot i \cdot (i \cdot \sqrt{16}) \cdot \sqrt{32}$

Step 10

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

$3 i \cdot (i \cdot \sqrt{4^{2}}) \cdot \sqrt{32}$

Step 11

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

$3 i \cdot (i \cdot 4) \cdot \sqrt{32}$

Step 12

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

$3 i \cdot (4 \cdot i) \cdot \sqrt{32}$

Step 13

Multiply $3 i (4 i)$ .

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Step 13.1

Multiply $4$ by $3$ .

$12 i i \cdot \sqrt{32}$

Step 13.2

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

$12 (i^{1} i) \cdot \sqrt{32}$

Step 13.3

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

$12 (i^{1} i^{1}) \cdot \sqrt{32}$

Step 13.4

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

$12 i^{1 + 1} \cdot \sqrt{32}$

Step 13.5

Add $1$ and $1$ .

$12 i^{2} \cdot \sqrt{32}$

Step 14

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

$12 \cdot - 1 \cdot \sqrt{32}$

Step 15

Multiply $12$ by $- 1$ .

$- 12 \cdot \sqrt{32}$

Step 16

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

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Step 16.1

Factor $16$ out of $32$ .

$- 12 \cdot \sqrt{16 (2)}$

Step 16.2

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

$- 12 \cdot \sqrt{4^{2} \cdot 2}$

Step 17

Pull terms out from under the radical.

$- 12 \cdot (4 \sqrt{2})$

Step 18

Multiply $4$ by $- 12$ .

$- 48 \sqrt{2}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$- 48 \sqrt{2}$

Decimal Form:

$- 67.88225099 \dots$