Simplify (4+4i)/(1+i)

$\frac{4 + 4 i}{1 + i}$

Multiply the numerator and denominator of $\frac{4 + 4 i}{1 + 1 i}$ by the conjugate of $1 + 1 i$ to make the denominator real.

$\frac{4 + 4 i}{1 + 1 i} \cdot \frac{1 - i}{1 - i}$

Multiply.

Combine.

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

Simplify the numerator.

Expand $(4 + 4 i) (1 - i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $1$ .

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

Multiply $- 1$ by $4$ .

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

Multiply $4$ by $1$ .

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

Multiply $4 i (- i)$ .

Multiply $- 1$ by $4$ .

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

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

$\frac{4 - 4 i + 4 i - 4 (i^{1} i)}{(1 + 1 i) (1 - i)}$

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

$\frac{4 - 4 i + 4 i - 4 (i^{1} i^{1})}{(1 + 1 i) (1 - i)}$

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

$\frac{4 - 4 i + 4 i - 4 i^{1 + 1}}{(1 + 1 i) (1 - i)}$

Add $1$ and $1$ .

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

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

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

Multiply $- 4$ by $- 1$ .

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

Add $4$ and $4$ .

$\frac{8 - 4 i + 4 i}{(1 + 1 i) (1 - i)}$

Add $- 4 i$ and $4 i$ .

$\frac{8 + 0}{(1 + 1 i) (1 - i)}$

Add $8$ and $0$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify.

Multiply $1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $1$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

$\frac{8}{1 - 1 i + 1 i - i^{1 + 1}}$

Add $1$ and $1$ .

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

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

$\frac{8}{1 + 0 - i^{2}}$

Add $1$ and $0$ .

$\frac{8}{1 - i^{2}}$

Simplify each term.

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

$\frac{8}{1 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{8}{1 + 1}$

Add $1$ and $1$ .

$\frac{8}{2}$

Divide $8$ by $2$ .

$4$

Simplify (4+4i)/(1+i)

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