Simplify (3+i)/(4+i)

$\frac{3 + i}{4 + i}$

Multiply the numerator and denominator of $\frac{3 + i}{4 + 1 i}$ by the conjugate of $4 + 1 i$ to make the denominator real.

$\frac{3 + i}{4 + 1 i} \cdot \frac{4 - i}{4 - i}$

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $4$ .

$\frac{12 + 3 (- i) + i \cdot 4 + i (- i)}{(4 + 1 i) (4 - i)}$

Multiply $- 1$ by $3$ .

$\frac{12 - 3 i + i \cdot 4 + i (- i)}{(4 + 1 i) (4 - i)}$

Move $4$ to the left of $i$ .

$\frac{12 - 3 i + 4 \cdot i + i (- i)}{(4 + 1 i) (4 - i)}$

Multiply $i (- i)$ .

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

$\frac{12 - 3 i + 4 i - (i^{1} i)}{(4 + 1 i) (4 - i)}$

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

$\frac{12 - 3 i + 4 i - (i^{1} i^{1})}{(4 + 1 i) (4 - i)}$

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

$\frac{12 - 3 i + 4 i - i^{1 + 1}}{(4 + 1 i) (4 - i)}$

Add $1$ and $1$ .

$\frac{12 - 3 i + 4 i - i^{2}}{(4 + 1 i) (4 - i)}$

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

$\frac{12 - 3 i + 4 i - - 1}{(4 + 1 i) (4 - i)}$

Multiply $- 1$ by $- 1$ .

$\frac{12 - 3 i + 4 i + 1}{(4 + 1 i) (4 - i)}$

Add $12$ and $1$ .

$\frac{13 - 3 i + 4 i}{(4 + 1 i) (4 - i)}$

Add $- 3 i$ and $4 i$ .

$\frac{13 + i}{(4 + 1 i) (4 - i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{13 + i}{4 (4 - i) + 1 i (4 - i)}$

Apply the distributive property.

$\frac{13 + i}{4 \cdot 4 + 4 (- i) + 1 i (4 - i)}$

Apply the distributive property.

$\frac{13 + i}{4 \cdot 4 + 4 (- i) + 1 i \cdot 4 + 1 i (- i)}$

Simplify.

Multiply $4$ by $4$ .

$\frac{13 + i}{16 + 4 (- i) + 1 i \cdot 4 + 1 i (- i)}$

Multiply $- 1$ by $4$ .

$\frac{13 + i}{16 - 4 i + 1 i \cdot 4 + 1 i (- i)}$

Multiply $4$ by $1$ .

$\frac{13 + i}{16 - 4 i + 4 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{13 + i}{16 - 4 i + 4 i - i i}$

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

$\frac{13 + i}{16 - 4 i + 4 i - (i^{1} i)}$

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

$\frac{13 + i}{16 - 4 i + 4 i - (i^{1} i^{1})}$

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

$\frac{13 + i}{16 - 4 i + 4 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{13 + i}{16 - 4 i + 4 i - i^{2}}$

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

$\frac{13 + i}{16 + 0 - i^{2}}$

Add $16$ and $0$ .

$\frac{13 + i}{16 - i^{2}}$

Simplify each term.

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

$\frac{13 + i}{16 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{13 + i}{16 + 1}$

Add $16$ and $1$ .

$\frac{13 + i}{17}$

Split the fraction $\frac{13 + i}{17}$ into two fractions.

$\frac{13}{17} + \frac{i}{17}$

Simplify (3+i)/(4+i)

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