Simplify (2+i)/(3+i)

$\frac{2 + i}{3 + i}$

Multiply the numerator and denominator of $\frac{2 + i}{3 + 1 i}$ by the conjugate of $3 + 1 i$ to make the denominator real.

$\frac{2 + i}{3 + 1 i} \cdot \frac{3 - i}{3 - i}$

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- i) + i (3 - i)}{(3 + 1 i) (3 - i)}$

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- i) + i \cdot 3 + i (- i)}{(3 + 1 i) (3 - i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $3$ .

$\frac{6 + 2 (- i) + i \cdot 3 + i (- i)}{(3 + 1 i) (3 - i)}$

Multiply $- 1$ by $2$ .

$\frac{6 - 2 i + i \cdot 3 + i (- i)}{(3 + 1 i) (3 - i)}$

Move $3$ to the left of $i$ .

$\frac{6 - 2 i + 3 \cdot i + i (- i)}{(3 + 1 i) (3 - i)}$

Multiply $i (- i)$ .

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

$\frac{6 - 2 i + 3 i - (i^{1} i)}{(3 + 1 i) (3 - i)}$

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

$\frac{6 - 2 i + 3 i - (i^{1} i^{1})}{(3 + 1 i) (3 - i)}$

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

$\frac{6 - 2 i + 3 i - i^{1 + 1}}{(3 + 1 i) (3 - i)}$

Add $1$ and $1$ .

$\frac{6 - 2 i + 3 i - i^{2}}{(3 + 1 i) (3 - i)}$

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

$\frac{6 - 2 i + 3 i - - 1}{(3 + 1 i) (3 - i)}$

Multiply $- 1$ by $- 1$ .

$\frac{6 - 2 i + 3 i + 1}{(3 + 1 i) (3 - i)}$

Add $6$ and $1$ .

$\frac{7 - 2 i + 3 i}{(3 + 1 i) (3 - i)}$

Add $- 2 i$ and $3 i$ .

$\frac{7 + i}{(3 + 1 i) (3 - i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{7 + i}{3 (3 - i) + 1 i (3 - i)}$

Apply the distributive property.

$\frac{7 + i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)}$

Apply the distributive property.

$\frac{7 + i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Simplify.

Multiply $3$ by $3$ .

$\frac{7 + i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Multiply $- 1$ by $3$ .

$\frac{7 + i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)}$

Multiply $3$ by $1$ .

$\frac{7 + i}{9 - 3 i + 3 i + 1 i (- i)}$

Multiply $- 1$ by $1$ .

$\frac{7 + i}{9 - 3 i + 3 i - i i}$

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

$\frac{7 + i}{9 - 3 i + 3 i - (i^{1} i)}$

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

$\frac{7 + i}{9 - 3 i + 3 i - (i^{1} i^{1})}$

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

$\frac{7 + i}{9 - 3 i + 3 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{7 + i}{9 - 3 i + 3 i - i^{2}}$

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

$\frac{7 + i}{9 + 0 - i^{2}}$

Add $9$ and $0$ .

$\frac{7 + i}{9 - i^{2}}$

Simplify each term.

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

$\frac{7 + i}{9 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{7 + i}{9 + 1}$

Add $9$ and $1$ .

$\frac{7 + i}{10}$

Split the fraction $\frac{7 + i}{10}$ into two fractions.

$\frac{7}{10} + \frac{i}{10}$

Simplify (2+i)/(3+i)

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