Algebra Examples

$\sqrt{- 6} \cdot \sqrt{- 12} \cdot \sqrt{- 5}$

Step 1

Rewrite $- 6$ as $- 1 (6)$ .

$\sqrt{- 1 (6)} \cdot \sqrt{- 12} \cdot \sqrt{- 5}$

Step 2

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$\sqrt{- 1} \cdot \sqrt{6} \cdot \sqrt{- 12} \cdot \sqrt{- 5}$

Step 3

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

$i \cdot \sqrt{6} \cdot \sqrt{- 12} \cdot \sqrt{- 5}$

Step 4

Rewrite $- 12$ as $- 1 (12)$ .

$i \cdot \sqrt{6} \cdot \sqrt{- 1 (12)} \cdot \sqrt{- 5}$

Step 5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$i \cdot \sqrt{6} \cdot (\sqrt{- 1} \cdot \sqrt{12}) \cdot \sqrt{- 5}$

Step 6

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

$i \cdot \sqrt{6} \cdot (i \cdot \sqrt{12}) \cdot \sqrt{- 5}$

Step 7

Rewrite $- 5$ as $- 1 (5)$ .

$i \cdot \sqrt{6} \cdot (i \cdot \sqrt{12}) \cdot \sqrt{- 1 (5)}$

Step 8

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$i \cdot \sqrt{6} \cdot (i \cdot \sqrt{12}) \cdot (\sqrt{- 1} \cdot \sqrt{5})$

Step 9

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

$i \cdot \sqrt{6} \cdot (i \cdot \sqrt{12}) \cdot (i \cdot \sqrt{5})$

Step 10

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

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Step 10.1

Factor $4$ out of $12$ .

$i \sqrt{6} \cdot (i \cdot \sqrt{4 (3)}) \cdot (i \cdot \sqrt{5})$

Step 10.2

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

$i \sqrt{6} \cdot (i \cdot \sqrt{2^{2} \cdot 3}) \cdot (i \cdot \sqrt{5})$

Step 11

Pull terms out from under the radical.

$i \sqrt{6} \cdot (i \cdot (| 2 | \sqrt{3})) \cdot (i \cdot \sqrt{5})$

Step 12

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

$i \sqrt{6} \cdot (i \cdot (2 \sqrt{3})) \cdot (i \cdot \sqrt{5})$

Step 13

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

$i \sqrt{6} \cdot (2 \cdot i \sqrt{3}) \cdot (i \cdot \sqrt{5})$

Step 14

Multiply $i \sqrt{6} (2 i \sqrt{3})$ .

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Step 14.1

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

$\sqrt{6} (2 (i^{1} i) \sqrt{3}) \cdot (i \cdot \sqrt{5})$

Step 14.2

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

$\sqrt{6} (2 (i^{1} i^{1}) \sqrt{3}) \cdot (i \cdot \sqrt{5})$

Step 14.3

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

$\sqrt{6} (2 i^{1 + 1} \sqrt{3}) \cdot (i \cdot \sqrt{5})$

Step 14.4

Add $1$ and $1$ .

$\sqrt{6} (2 i^{2} \sqrt{3}) \cdot (i \cdot \sqrt{5})$

Step 14.5

Combine using the product rule for radicals.

$2 i^{2} \sqrt{6 \cdot 3} \cdot (i \cdot \sqrt{5})$

Step 14.6

Multiply $6$ by $3$ .

$2 i^{2} \sqrt{18} \cdot (i \cdot \sqrt{5})$

Step 15

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

$2 \cdot - 1 \sqrt{18} \cdot (i \cdot \sqrt{5})$

Step 16

Multiply $2$ by $- 1$ .

$- 2 \sqrt{18} \cdot (i \cdot \sqrt{5})$

Step 17

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 17.1

Factor $9$ out of $18$ .

$- 2 \sqrt{9 (2)} \cdot (i \cdot \sqrt{5})$

Step 17.2

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

$- 2 \sqrt{3^{2} \cdot 2} \cdot (i \cdot \sqrt{5})$

Step 18

Pull terms out from under the radical.

$- 2 (| 3 | \sqrt{2}) \cdot (i \cdot \sqrt{5})$

Step 19

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$- 2 (3 \sqrt{2}) \cdot (i \cdot \sqrt{5})$

Step 20

Multiply $3$ by $- 2$ .

$- 6 \sqrt{2} \cdot (i \cdot \sqrt{5})$

Step 21

Multiply $- 6 \sqrt{2} (i \sqrt{5})$ .

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Step 21.1

Combine using the product rule for radicals.

$- 6 \sqrt{5 \cdot 2} i$

Step 21.2

Multiply $5$ by $2$ .

$- 6 \sqrt{10} i$