Evaluate (6- square root of -3)(1+ square root of -1)

$(6 - \sqrt{- 3}) (1 + \sqrt{- 1})$

Step 1

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

$(6 - \sqrt{- 1 (3)}) (1 + \sqrt{- 1})$

Step 2

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

$(6 - (\sqrt{- 1} \cdot \sqrt{3})) (1 + \sqrt{- 1})$

Step 3

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

$(6 - (i \cdot \sqrt{3})) (1 + \sqrt{- 1})$

Step 4

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

$(6 - (i \cdot \sqrt{3})) (1 + i)$

Step 5

Remove parentheses.

$(6 - i \sqrt{3}) (1 + i)$

Step 6

Expand $(6 - i \sqrt{3}) (1 + i)$ using the FOIL Method.

Apply the distributive property.

$6 (1 + i) - i \sqrt{3} (1 + i)$

Apply the distributive property.

$6 \cdot 1 + 6 i - i \sqrt{3} (1 + i)$

Apply the distributive property.

$6 \cdot 1 + 6 i - i \sqrt{3} \cdot 1 - i \sqrt{3} i$

Step 7

Simplify terms.

Simplify each term.

Multiply $6$ by $1$ .

$6 + 6 i - i \sqrt{3} \cdot 1 - i \sqrt{3} i$

Multiply $- 1$ by $1$ .

$6 + 6 i - i \sqrt{3} - i \sqrt{3} i$

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

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

$6 + 6 i - i \sqrt{3} - (i^{1} i) \sqrt{3}$

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

$6 + 6 i - i \sqrt{3} - (i^{1} i^{1}) \sqrt{3}$

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

$6 + 6 i - i \sqrt{3} - i^{1 + 1} \sqrt{3}$

Add $1$ and $1$ .

$6 + 6 i - i \sqrt{3} - i^{2} \sqrt{3}$

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

$6 + 6 i - i \sqrt{3} - - 1 \sqrt{3}$

Multiply $- 1$ by $- 1$ .

$6 + 6 i - i \sqrt{3} + 1 \sqrt{3}$

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

$6 + 6 i - i \sqrt{3} + \sqrt{3}$

Reorder $6 + 6 i - i \sqrt{3}$ and $\sqrt{3}$ .

$\sqrt{3} + 6 + 6 i - i \sqrt{3}$

Evaluate (6- square root of -3)(1+ square root of -1)

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