Algebra Examples

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

Step 1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify each term.

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Step 5.1

Rewrite $4$ as $2^{2}$ .

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

Step 5.2

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

$(3 + i \cdot 2) (4 + i)$

Step 5.3

Move $2$ to the left of $i$ .

$(3 + 2 i) (4 + i)$

Step 6

Expand $(3 + 2 i) (4 + i)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$3 (4 + i) + 2 i (4 + i)$

Step 6.2

Apply the distributive property.

$3 \cdot 4 + 3 i + 2 i (4 + i)$

Step 6.3

Apply the distributive property.

$3 \cdot 4 + 3 i + 2 i \cdot 4 + 2 i i$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $3$ by $4$ .

$12 + 3 i + 2 i \cdot 4 + 2 i i$

Step 7.1.2

Multiply $4$ by $2$ .

$12 + 3 i + 8 i + 2 i i$

Step 7.1.3

Multiply $2 i i$ .

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Step 7.1.3.1

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

$12 + 3 i + 8 i + 2 (i^{1} i)$

Step 7.1.3.2

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

$12 + 3 i + 8 i + 2 (i^{1} i^{1})$

Step 7.1.3.3

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

$12 + 3 i + 8 i + 2 i^{1 + 1}$

Step 7.1.3.4

Add $1$ and $1$ .

$12 + 3 i + 8 i + 2 i^{2}$

Step 7.1.4

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

$12 + 3 i + 8 i + 2 \cdot - 1$

Step 7.1.5

Multiply $2$ by $- 1$ .

$12 + 3 i + 8 i - 2$

Step 7.2

Subtract $2$ from $12$ .

$10 + 3 i + 8 i$

Step 7.3

Add $3 i$ and $8 i$ .

$10 + 11 i$