Basic Math Examples

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

Step 1

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Remove parentheses.

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

Step 6

Expand $(2 - i \sqrt{3}) (1 + i)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $2$ by $1$ .

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

Step 7.1.2

Multiply $- 1$ by $1$ .

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

Step 7.1.3

Multiply $- i \sqrt{3} i$ .

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Step 7.1.3.1

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

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

Step 7.1.3.2

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

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

Step 7.1.3.3

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

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

Step 7.1.3.4

Add $1$ and $1$ .

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

Step 7.1.4

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

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

Step 7.1.5

Multiply $- 1$ by $- 1$ .

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

Step 7.1.6

Multiply $\sqrt{3}$ by $1$ .

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

Step 7.2

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

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