Algebra Examples

$\sqrt{- 6} \cdot \sqrt{- 3} \cdot \sqrt{- 5}$

Step 1

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

$\sqrt{- 1 (6)} \cdot \sqrt{- 3} \cdot \sqrt{- 5}$

Step 2

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

$\sqrt{- 1} \cdot \sqrt{6} \cdot \sqrt{- 3} \cdot \sqrt{- 5}$

Step 3

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

$i \cdot \sqrt{6} \cdot \sqrt{- 3} \cdot \sqrt{- 5}$

Step 4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

Add $1$ and $1$ .

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

Step 10.5

Combine using the product rule for radicals.

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

Step 10.6

Multiply $3$ by $6$ .

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

Step 11

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

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

Step 12

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

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Step 12.1

Factor $9$ out of $18$ .

$- 1 \sqrt{9 (2)} \cdot (i \cdot \sqrt{5})$

Step 12.2

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

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

Step 13

Pull terms out from under the radical.

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

Step 14

Multiply $3$ by $- 1$ .

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

Step 15

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

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Step 15.1

Combine using the product rule for radicals.

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

Step 15.2

Multiply $5$ by $2$ .

$- 3 \sqrt{10} i$