Precalculus Examples

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

Step 1

Rewrite $- 2$ as $- 1 (2)$ .

$(\sqrt{- 1 (2)} + 2 i) (6 + 5 i)$

Step 2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

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

Step 3

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

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

Step 4

Expand $(i \sqrt{2} + 2 i) (6 + 5 i)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$i \sqrt{2} (6 + 5 i) + 2 i (6 + 5 i)$

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Move $6$ to the left of $i \sqrt{2}$ .

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

Step 5.1.2

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

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Step 5.1.2.1

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

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

Step 5.1.2.2

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

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

Step 5.1.2.3

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

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

Step 5.1.2.4

Add $1$ and $1$ .

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

Step 5.1.3

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

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

Step 5.1.4

Multiply $5$ by $- 1$ .

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

Step 5.1.5

Move $- 5$ to the left of $\sqrt{2}$ .

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

Step 5.1.6

Multiply $6$ by $2$ .

$6 i \sqrt{2} - 5 \sqrt{2} + 12 i + 2 i (5 i)$

Step 5.1.7

Multiply $2 i (5 i)$ .

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Step 5.1.7.1

Multiply $5$ by $2$ .

$6 i \sqrt{2} - 5 \sqrt{2} + 12 i + 10 i i$

Step 5.1.7.2

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

$6 i \sqrt{2} - 5 \sqrt{2} + 12 i + 10 (i^{1} i)$

Step 5.1.7.3

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

$6 i \sqrt{2} - 5 \sqrt{2} + 12 i + 10 (i^{1} i^{1})$

Step 5.1.7.4

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

$6 i \sqrt{2} - 5 \sqrt{2} + 12 i + 10 i^{1 + 1}$

Step 5.1.7.5

Add $1$ and $1$ .

$6 i \sqrt{2} - 5 \sqrt{2} + 12 i + 10 i^{2}$

Step 5.1.8

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

$6 i \sqrt{2} - 5 \sqrt{2} + 12 i + 10 \cdot - 1$

Step 5.1.9

Multiply $10$ by $- 1$ .

$6 i \sqrt{2} - 5 \sqrt{2} + 12 i - 10$

Step 5.2

Simplify the expression.

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Step 5.2.1

Reorder $6 i \sqrt{2}$ and $- 5 \sqrt{2}$ .

$- 5 \sqrt{2} + 6 i \sqrt{2} + 12 i - 10$

Step 5.2.2

Reorder $- 5 \sqrt{2} + 6 i \sqrt{2} + 12 i$ and $- 10$ .

$- 10 - 5 \sqrt{2} + 6 i \sqrt{2} + 12 i$