Precalculus Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Move $- 2$ to the left of $\sqrt{3}$ .

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

Step 2.1.2

Multiply $- 2$ by $- 1$ .

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

Step 2.1.3

Multiply $- i (- 2 i)$ .

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Step 2.1.3.1

Multiply $- 2$ by $- 1$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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

Step 2.1.3.5

Add $1$ and $1$ .

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

Step 2.1.4

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

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

Step 2.1.5

Multiply $2$ by $- 1$ .

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

Step 2.2

Simplify the expression.

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Step 2.2.1

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

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

Step 2.2.2

Move parentheses.

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