Algebra Examples

$\sqrt{- 8} \cdot \sqrt{24}$

Step 1

Rewrite $- 8$ as $- 1 (8)$ .

$\sqrt{- 1 (8)} \cdot \sqrt{24}$

Step 2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$\sqrt{- 1} \cdot \sqrt{8} \cdot \sqrt{24}$

Step 3

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

$i \cdot \sqrt{8} \cdot \sqrt{24}$

Step 4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.1

Factor $4$ out of $8$ .

$i \cdot \sqrt{4 (2)} \cdot \sqrt{24}$

Step 4.2

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

$i \cdot \sqrt{2^{2} \cdot 2} \cdot \sqrt{24}$

Step 5

Pull terms out from under the radical.

$i \cdot (| 2 | \sqrt{2}) \cdot \sqrt{24}$

Step 6

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 7

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

$2 \cdot i \sqrt{2} \cdot \sqrt{24}$

Step 8

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

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Step 8.1

Factor $4$ out of $24$ .

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

Step 8.2

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

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

Step 9

Pull terms out from under the radical.

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

Step 10

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 11

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

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Step 11.1

Multiply $2$ by $2$ .

$4 i \sqrt{2} \sqrt{6}$

Step 11.2

Combine using the product rule for radicals.

$4 i \sqrt{6 \cdot 2}$

Step 11.3

Multiply $6$ by $2$ .

$4 i \sqrt{12}$

Step 12

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

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Step 12.1

Factor $4$ out of $12$ .

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

Step 12.2

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

$4 i \sqrt{2^{2} \cdot 3}$

Step 13

Pull terms out from under the radical.

$4 i (| 2 | \sqrt{3})$

Step 14

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 15

Multiply $2$ by $4$ .

$8 i \sqrt{3}$