Algebra Examples

$\sqrt{- 6} (\sqrt{- 8} - 2 \sqrt{2})$

Step 1

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

$\sqrt{- 1 (6)} (\sqrt{- 8} - 2 \sqrt{2})$

Step 2

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

$\sqrt{- 1} \cdot \sqrt{6} (\sqrt{- 8} - 2 \sqrt{2})$

Step 3

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

$i \cdot \sqrt{6} (\sqrt{- 8} - 2 \sqrt{2})$

Step 4

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

$i \cdot \sqrt{6} (\sqrt{- 1 (8)} - 2 \sqrt{2})$

Step 5

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

$i \cdot \sqrt{6} (\sqrt{- 1} \cdot \sqrt{8} - 2 \sqrt{2})$

Step 6

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

$i \cdot \sqrt{6} (i \cdot \sqrt{8} - 2 \sqrt{2})$

Step 7

Simplify each term.

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Step 7.1

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

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Step 7.1.1

Factor $4$ out of $8$ .

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

Step 7.1.2

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

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

Step 7.2

Pull terms out from under the radical.

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

Step 7.3

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

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

Step 8

Apply the distributive property.

$i \sqrt{6} (2 i \sqrt{2}) + i \sqrt{6} (- 2 \sqrt{2})$

Step 9

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

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Step 9.1

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

$\sqrt{6} (2 (i^{1} i) \sqrt{2}) + i \sqrt{6} (- 2 \sqrt{2})$

Step 9.2

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

$\sqrt{6} (2 (i^{1} i^{1}) \sqrt{2}) + i \sqrt{6} (- 2 \sqrt{2})$

Step 9.3

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

$\sqrt{6} (2 i^{1 + 1} \sqrt{2}) + i \sqrt{6} (- 2 \sqrt{2})$

Step 9.4

Add $1$ and $1$ .

$\sqrt{6} (2 i^{2} \sqrt{2}) + i \sqrt{6} (- 2 \sqrt{2})$

Step 9.5

Combine using the product rule for radicals.

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

Step 9.6

Multiply $6$ by $2$ .

$2 i^{2} \sqrt{12} + i \sqrt{6} (- 2 \sqrt{2})$

Step 10

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

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Step 10.1

Combine using the product rule for radicals.

$2 i^{2} \sqrt{12} + i (- 2 \sqrt{6 \cdot 2})$

Step 10.2

Multiply $6$ by $2$ .

$2 i^{2} \sqrt{12} + i (- 2 \sqrt{12})$

Step 11

Simplify each term.

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Step 11.1

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

$2 \cdot - 1 \sqrt{12} + i (- 2 \sqrt{12})$

Step 11.2

Multiply $2$ by $- 1$ .

$- 2 \sqrt{12} + i (- 2 \sqrt{12})$

Step 11.3

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

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Step 11.3.1

Factor $4$ out of $12$ .

$- 2 \sqrt{4 (3)} + i (- 2 \sqrt{12})$

Step 11.3.2

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

$- 2 \sqrt{2^{2} \cdot 3} + i (- 2 \sqrt{12})$

Step 11.4

Pull terms out from under the radical.

$- 2 (2 \sqrt{3}) + i (- 2 \sqrt{12})$

Step 11.5

Multiply $2$ by $- 2$ .

$- 4 \sqrt{3} + i (- 2 \sqrt{12})$

Step 11.6

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

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Step 11.6.1

Factor $4$ out of $12$ .

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

Step 11.6.2

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

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

Step 11.7

Pull terms out from under the radical.

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

Step 11.8

Multiply $2$ by $- 2$ .

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

Step 12

Reorder factors in $- 4 \sqrt{3} + i (- 4 \sqrt{3})$ .

$- 4 \sqrt{3} - 4 i \sqrt{3}$