Trigonometry Examples

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

Step 1

Expand $(\sqrt{2} + 3 i) (\sqrt{2} - 3 i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine using the product rule for radicals.

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.1.3

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

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

Step 2.1.4

Pull terms out from under the radical, assuming positive real numbers.

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

Step 2.1.5

Multiply $3 i (- 3 i)$ .

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Step 2.1.5.1

Multiply $- 3$ by $3$ .

$2 + \sqrt{2} (- 3 i) + 3 i \sqrt{2} - 9 i i$

Step 2.1.5.2

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

$2 + \sqrt{2} (- 3 i) + 3 i \sqrt{2} - 9 (i^{1} i)$

Step 2.1.5.3

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

$2 + \sqrt{2} (- 3 i) + 3 i \sqrt{2} - 9 (i^{1} i^{1})$

Step 2.1.5.4

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

$2 + \sqrt{2} (- 3 i) + 3 i \sqrt{2} - 9 i^{1 + 1}$

Step 2.1.5.5

Add $1$ and $1$ .

$2 + \sqrt{2} (- 3 i) + 3 i \sqrt{2} - 9 i^{2}$

Step 2.1.6

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

$2 + \sqrt{2} (- 3 i) + 3 i \sqrt{2} - 9 \cdot - 1$

Step 2.1.7

Multiply $- 9$ by $- 1$ .

$2 + \sqrt{2} (- 3 i) + 3 i \sqrt{2} + 9$

Step 2.2

Add $2$ and $9$ .

$\sqrt{2} (- 3 i) + 3 i \sqrt{2} + 11$

Step 2.3

Reorder the factors of $\sqrt{2} (- 3 i)$ .

$- 3 i \sqrt{2} + 3 i \sqrt{2} + 11$

Step 2.4

Add $- 3 i \sqrt{2}$ and $3 i \sqrt{2}$ .

$0 + 11$

Step 2.5

Add $0$ and $11$ .

$11$