Simplify (3+2i)/(5-i)

$\frac{3 + 2 i}{5 - i}$

Multiply the numerator and denominator of $\frac{3 + 2 i}{5 - i}$ by the conjugate of $5 - i$ to make the denominator real.

$\frac{3 + 2 i}{5 - i} \cdot \frac{5 + i}{5 + i}$

Multiply.

Combine.

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

Simplify the numerator.

Expand $(3 + 2 i) (5 + i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$\frac{3 \cdot 5 + 3 i + 2 i (5 + i)}{(5 - i) (5 + i)}$

Apply the distributive property.

$\frac{3 \cdot 5 + 3 i + 2 i \cdot 5 + 2 i i}{(5 - i) (5 + i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $5$ .

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

Multiply $5$ by $2$ .

$\frac{15 + 3 i + 10 i + 2 i i}{(5 - i) (5 + i)}$

Multiply $2 i i$ .

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

$\frac{15 + 3 i + 10 i + 2 (i^{1} i)}{(5 - i) (5 + i)}$

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

$\frac{15 + 3 i + 10 i + 2 (i^{1} i^{1})}{(5 - i) (5 + i)}$

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

$\frac{15 + 3 i + 10 i + 2 i^{1 + 1}}{(5 - i) (5 + i)}$

Add $1$ and $1$ .

$\frac{15 + 3 i + 10 i + 2 i^{2}}{(5 - i) (5 + i)}$

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

$\frac{15 + 3 i + 10 i + 2 \cdot - 1}{(5 - i) (5 + i)}$

Multiply $2$ by $- 1$ .

$\frac{15 + 3 i + 10 i - 2}{(5 - i) (5 + i)}$

Subtract $2$ from $15$ .

$\frac{13 + 3 i + 10 i}{(5 - i) (5 + i)}$

Add $3 i$ and $10 i$ .

$\frac{13 + 13 i}{(5 - i) (5 + i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{13 + 13 i}{5 (5 + i) - i (5 + i)}$

Apply the distributive property.

$\frac{13 + 13 i}{5 \cdot 5 + 5 i - i (5 + i)}$

Apply the distributive property.

$\frac{13 + 13 i}{5 \cdot 5 + 5 i - i \cdot 5 - i i}$

Simplify.

Multiply $5$ by $5$ .

$\frac{13 + 13 i}{25 + 5 i - i \cdot 5 - i i}$

Multiply $5$ by $- 1$ .

$\frac{13 + 13 i}{25 + 5 i - 5 i - i i}$

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

$\frac{13 + 13 i}{25 + 5 i - 5 i - (i^{1} i)}$

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

$\frac{13 + 13 i}{25 + 5 i - 5 i - (i^{1} i^{1})}$

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

$\frac{13 + 13 i}{25 + 5 i - 5 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{13 + 13 i}{25 + 5 i - 5 i - i^{2}}$

Subtract $5 i$ from $5 i$ .

$\frac{13 + 13 i}{25 + 0 - i^{2}}$

Add $25$ and $0$ .

$\frac{13 + 13 i}{25 - i^{2}}$

Simplify each term.

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

$\frac{13 + 13 i}{25 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{13 + 13 i}{25 + 1}$

Add $25$ and $1$ .

$\frac{13 + 13 i}{26}$

Cancel the common factor of $13 + 13 i$ and $26$ .

Factor $13$ out of $13$ .

$\frac{13 \cdot 1 + 13 i}{26}$

Factor $13$ out of $13 i$ .

$\frac{13 \cdot 1 + 13 (i)}{26}$

Factor $13$ out of $13 \cdot 1 + 13 (i)$ .

$\frac{13 \cdot (1 + i)}{26}$

Cancel the common factors.

Factor $13$ out of $26$ .

$\frac{13 \cdot (1 + i)}{13 (2)}$

Cancel the common factor.

$\frac{13 \cdot (1 + i)}{13 \cdot 2}$

Rewrite the expression.

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

Split the fraction $\frac{1 + i}{2}$ into two fractions.

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

Simplify (3+2i)/(5-i)

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