Simplify (2+ square root of -8)/(1+i)

$\frac{2 + \sqrt{- 8}}{1 + i}$

Pull out imaginary unit $i$ .

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{2 + \sqrt{- 1 (8)}}{1 + i}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$\frac{2 + \sqrt{- 1} \cdot \sqrt{8}}{1 + i}$

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

$\frac{2 + i \cdot \sqrt{8}}{1 + i}$

Multiply the numerator and denominator of $\frac{2 + i \cdot \sqrt{8}}{1 + 1 i}$ by the conjugate of $1 + 1 i$ to make the denominator real.

$\frac{2 + i \cdot \sqrt{8}}{1 + 1 i} \cdot \frac{1 - i}{1 - i}$

Multiply.

Combine.

$\frac{(2 + i \cdot \sqrt{8}) (1 - i)}{(1 + 1 i) (1 - i)}$

Simplify the numerator.

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Multiply $2$ by $1$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $1$ .

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

Multiply $2 i \sqrt{2} (- i)$ .

Multiply $- 1$ by $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 2$ by $- 1$ .

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

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

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

Simplify the denominator.

Expand $(1 + 1 i) (1 - i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Add $- 1 i$ and $1 i$ .

$\frac{2 \sqrt{2} + 2 - 2 i + 2 i \sqrt{2}}{1 + 0 - i^{2}}$

Add $1$ and $0$ .

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

Simplify each term.

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

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

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

Cancel the common factor of $2 \sqrt{2} + 2 - 2 i + 2 i \sqrt{2}$ and $2$ .

Factor $2$ out of $2 \sqrt{2}$ .

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

Factor $2$ out of $2$ .

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

Factor $2$ out of $- 2 i$ .

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

Factor $2$ out of $2 (1) + 2 (- i)$ .

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

Factor $2$ out of $2 i \sqrt{2}$ .

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

Factor $2$ out of $2 (1 - i) + 2 (i \sqrt{2})$ .

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

Factor $2$ out of $2 (\sqrt{2}) + 2 (1 - i + i \sqrt{2})$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $\sqrt{2} + 1 - i + i \sqrt{2}$ by $1$ .

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

Simplify (2+ square root of -8)/(1+i)

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