Algebra Examples

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

Step 1

Rewrite ${(\sqrt{3} - 2 i)}^{2}$ as $(\sqrt{3} - 2 i) (\sqrt{3} - 2 i)$ .

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine using the product rule for radicals.

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

Step 3.1.2

Multiply $3$ by $3$ .

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

Step 3.1.3

Rewrite $9$ as $3^{2}$ .

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

Step 3.1.4

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

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

Step 3.1.5

Multiply $- 2 i (- 2 i)$ .

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Step 3.1.5.1

Multiply $- 2$ by $- 2$ .

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

Step 3.1.5.2

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

$3 + \sqrt{3} (- 2 i) - 2 i \sqrt{3} + 4 (i^{1} i)$

Step 3.1.5.3

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

$3 + \sqrt{3} (- 2 i) - 2 i \sqrt{3} + 4 (i^{1} i^{1})$

Step 3.1.5.4

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

$3 + \sqrt{3} (- 2 i) - 2 i \sqrt{3} + 4 i^{1 + 1}$

Step 3.1.5.5

Add $1$ and $1$ .

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

Step 3.1.6

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

$3 + \sqrt{3} (- 2 i) - 2 i \sqrt{3} + 4 \cdot - 1$

Step 3.1.7

Multiply $4$ by $- 1$ .

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

Step 3.2

Subtract $4$ from $3$ .

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

Step 3.3

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

$- 2 i \sqrt{3} - 2 i \sqrt{3} - 1$

Step 3.4

Subtract $2 i \sqrt{3}$ from $- 2 i \sqrt{3}$ .

$- 4 i \sqrt{3} - 1$

Step 4

Reorder $- 4 i \sqrt{3}$ and $- 1$ .

$- 1 - 4 i \sqrt{3}$