Simplify (1+i)/(3-i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $1$ .

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

Multiply $i$ by $1$ .

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

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

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

Multiply $i i$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Subtract $1$ from $3$ .

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

Add $i$ and $3 i$ .

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

Simplify the denominator.

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

Apply the distributive property.

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

Apply the distributive property.

$\frac{2 + 4 i}{3 \cdot 3 + 3 i - i (3 + i)}$

Apply the distributive property.

$\frac{2 + 4 i}{3 \cdot 3 + 3 i - i \cdot 3 - i i}$

Simplify.

Multiply $3$ by $3$ .

$\frac{2 + 4 i}{9 + 3 i - i \cdot 3 - i i}$

Multiply $3$ by $- 1$ .

$\frac{2 + 4 i}{9 + 3 i - 3 i - i i}$

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

$\frac{2 + 4 i}{9 + 3 i - 3 i - (i^{1} i)}$

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

$\frac{2 + 4 i}{9 + 3 i - 3 i - (i^{1} i^{1})}$

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

$\frac{2 + 4 i}{9 + 3 i - 3 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{2 + 4 i}{9 + 3 i - 3 i - i^{2}}$

Subtract $3 i$ from $3 i$ .

$\frac{2 + 4 i}{9 + 0 - i^{2}}$

Add $9$ and $0$ .

$\frac{2 + 4 i}{9 - i^{2}}$

Simplify each term.

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

$\frac{2 + 4 i}{9 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{2 + 4 i}{9 + 1}$

Add $9$ and $1$ .

$\frac{2 + 4 i}{10}$

Step 3

Cancel the common factor of $2 + 4 i$ and $10$ .

Factor $2$ out of $2$ .

$\frac{2 (1) + 4 i}{10}$

Factor $2$ out of $4 i$ .

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

Factor $2$ out of $2 (1) + 2 (2 i)$ .

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

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{2 (1 + 2 i)}{2 \cdot 5}$

Cancel the common factor.

$\frac{2 (1 + 2 i)}{2 \cdot 5}$

Rewrite the expression.

$\frac{1 + 2 i}{5}$

Step 4

Split the fraction $\frac{1 + 2 i}{5}$ into two fractions.

$\frac{1}{5} + \frac{2 i}{5}$

Simplify (1+i)/(3-i)

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