Trigonometry Examples

$(1 + i \sqrt{3}) (1 + i)$

Step 1

Expand $(1 + i \sqrt{3}) (1 + i)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$1 (1 + i) + i \sqrt{3} (1 + i)$

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

$1 \cdot 1 + 1 i + i \sqrt{3} \cdot 1 + i \sqrt{3} i$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $1$ by $1$ .

$1 + 1 i + i \sqrt{3} \cdot 1 + i \sqrt{3} i$

Step 2.1.2

Multiply $i$ by $1$ .

$1 + i + i \sqrt{3} \cdot 1 + i \sqrt{3} i$

Step 2.1.3

Multiply $i$ by $1$ .

$1 + i + i \sqrt{3} + i \sqrt{3} i$

Step 2.1.4

Multiply $i \sqrt{3} i$ .

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Step 2.1.4.1

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

$1 + i + i \sqrt{3} + i^{1} i \sqrt{3}$

Step 2.1.4.2

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

$1 + i + i \sqrt{3} + i^{1} i^{1} \sqrt{3}$

Step 2.1.4.3

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

$1 + i + i \sqrt{3} + i^{1 + 1} \sqrt{3}$

Step 2.1.4.4

Add $1$ and $1$ .

$1 + i + i \sqrt{3} + i^{2} \sqrt{3}$

Step 2.1.5

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

$1 + i + i \sqrt{3} - 1 \sqrt{3}$

Step 2.1.6

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$1 + i + i \sqrt{3} - \sqrt{3}$

Step 2.2

Reorder $1 + i + i \sqrt{3}$ and $- \sqrt{3}$ .

$- \sqrt{3} + 1 + i + i \sqrt{3}$