Simplify (5+5i)÷(3+i)

$(5 + 5 i) \div (3 + i)$

Rewrite the division as a fraction.

$\frac{5 + 5 i}{3 + i}$

Multiply the numerator and denominator of $\frac{5 + 5 i}{3 + 1 i}$ by the conjugate of $3 + 1 i$ to make the denominator real.

$\frac{5 + 5 i}{3 + 1 i} \cdot \frac{3 - i}{3 - i}$

Multiply.

Combine.

$\frac{(5 + 5 i) (3 - i)}{(3 + 1 i) (3 - i)}$

Simplify the numerator.

Expand $(5 + 5 i) (3 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 (3 - i) + 5 i (3 - i)}{(3 + 1 i) (3 - i)}$

Apply the distributive property.

$\frac{5 \cdot 3 + 5 (- i) + 5 i (3 - i)}{(3 + 1 i) (3 - i)}$

Apply the distributive property.

$\frac{5 \cdot 3 + 5 (- i) + 5 i \cdot 3 + 5 i (- i)}{(3 + 1 i) (3 - i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $3$ .

$\frac{15 + 5 (- i) + 5 i \cdot 3 + 5 i (- i)}{(3 + 1 i) (3 - i)}$

Multiply $- 1$ by $5$ .

$\frac{15 - 5 i + 5 i \cdot 3 + 5 i (- i)}{(3 + 1 i) (3 - i)}$

Multiply $3$ by $5$ .

$\frac{15 - 5 i + 15 i + 5 i (- i)}{(3 + 1 i) (3 - i)}$

Multiply $5 i (- i)$ .

Multiply $- 1$ by $5$ .

$\frac{15 - 5 i + 15 i - 5 i i}{(3 + 1 i) (3 - i)}$

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

$\frac{15 - 5 i + 15 i - 5 (i^{1} i)}{(3 + 1 i) (3 - i)}$

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

$\frac{15 - 5 i + 15 i - 5 (i^{1} i^{1})}{(3 + 1 i) (3 - i)}$

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

$\frac{15 - 5 i + 15 i - 5 i^{1 + 1}}{(3 + 1 i) (3 - i)}$

Add $1$ and $1$ .

$\frac{15 - 5 i + 15 i - 5 i^{2}}{(3 + 1 i) (3 - i)}$

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

$\frac{15 - 5 i + 15 i - 5 \cdot - 1}{(3 + 1 i) (3 - i)}$

Multiply $- 5$ by $- 1$ .

$\frac{15 - 5 i + 15 i + 5}{(3 + 1 i) (3 - i)}$

Add $15$ and $5$ .

$\frac{20 - 5 i + 15 i}{(3 + 1 i) (3 - i)}$

Add $- 5 i$ and $15 i$ .

$\frac{20 + 10 i}{(3 + 1 i) (3 - i)}$

Simplify the denominator.

Expand $(3 + 1 i) (3 - i)$ using the FOIL Method.

Apply the distributive property.

$\frac{20 + 10 i}{3 (3 - i) + 1 i (3 - i)}$

Apply the distributive property.

$\frac{20 + 10 i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)}$

Apply the distributive property.

$\frac{20 + 10 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Simplify.

Multiply $3$ by $3$ .

$\frac{20 + 10 i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Multiply $- 1$ by $3$ .

$\frac{20 + 10 i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)}$

Multiply $3$ by $1$ .

$\frac{20 + 10 i}{9 - 3 i + 3 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{20 + 10 i}{9 - 3 i + 3 i - i i}$

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

$\frac{20 + 10 i}{9 - 3 i + 3 i - (i^{1} i)}$

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

$\frac{20 + 10 i}{9 - 3 i + 3 i - (i^{1} i^{1})}$

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

$\frac{20 + 10 i}{9 - 3 i + 3 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{20 + 10 i}{9 - 3 i + 3 i - i^{2}}$

Add $- 3 i$ and $3 i$ .

$\frac{20 + 10 i}{9 + 0 - i^{2}}$

Add $9$ and $0$ .

$\frac{20 + 10 i}{9 - i^{2}}$

Simplify each term.

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

$\frac{20 + 10 i}{9 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{20 + 10 i}{9 + 1}$

Add $9$ and $1$ .

$\frac{20 + 10 i}{10}$

Cancel the common factor of $20 + 10 i$ and $10$ .

Factor $10$ out of $20$ .

$\frac{10 \cdot 2 + 10 i}{10}$

Factor $10$ out of $10 i$ .

$\frac{10 \cdot 2 + 10 (i)}{10}$

Factor $10$ out of $10 (2) + 10 (i)$ .

$\frac{10 (2 + i)}{10}$

Cancel the common factors.

Factor $10$ out of $10$ .

$\frac{10 (2 + i)}{10 (1)}$

Cancel the common factor.

$\frac{10 (2 + i)}{10 \cdot 1}$

Rewrite the expression.

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

Divide $2 + i$ by $1$ .

$2 + i$

Simplify (5+5i)÷(3+i)

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