Evaluate (1+i)/(3+4i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(1 + i) (3 - 4 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $1$ .

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

Multiply $- 4 i$ by $1$ .

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

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

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

Multiply $i (- 4 i)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $- 4$ by $- 1$ .

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

Add $3$ and $4$ .

$\frac{7 - 4 i + 3 i}{(3 + 4 i) (3 - 4 i)}$

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

$\frac{7 - i}{(3 + 4 i) (3 - 4 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{7 - i}{3 (3 - 4 i) + 4 i (3 - 4 i)}$

Apply the distributive property.

$\frac{7 - i}{3 \cdot 3 + 3 (- 4 i) + 4 i (3 - 4 i)}$

Apply the distributive property.

$\frac{7 - i}{3 \cdot 3 + 3 (- 4 i) + 4 i \cdot 3 + 4 i (- 4 i)}$

Simplify.

Multiply $3$ by $3$ .

$\frac{7 - i}{9 + 3 (- 4 i) + 4 i \cdot 3 + 4 i (- 4 i)}$

Multiply $- 4$ by $3$ .

$\frac{7 - i}{9 - 12 i + 4 i \cdot 3 + 4 i (- 4 i)}$

Multiply $3$ by $4$ .

$\frac{7 - i}{9 - 12 i + 12 i + 4 i (- 4 i)}$

Multiply $- 4$ by $4$ .

$\frac{7 - i}{9 - 12 i + 12 i - 16 i i}$

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

$\frac{7 - i}{9 - 12 i + 12 i - 16 (i^{1} i)}$

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

$\frac{7 - i}{9 - 12 i + 12 i - 16 (i^{1} i^{1})}$

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

$\frac{7 - i}{9 - 12 i + 12 i - 16 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{7 - i}{9 - 12 i + 12 i - 16 i^{2}}$

Add $- 12 i$ and $12 i$ .

$\frac{7 - i}{9 + 0 - 16 i^{2}}$

Add $9$ and $0$ .

$\frac{7 - i}{9 - 16 i^{2}}$

Simplify each term.

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

$\frac{7 - i}{9 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$\frac{7 - i}{9 + 16}$

Add $9$ and $16$ .

$\frac{7 - i}{25}$

Step 3

Split the fraction $\frac{7 - i}{25}$ into two fractions.

$\frac{7}{25} + \frac{- i}{25}$

Step 4

Move the negative in front of the fraction.

$\frac{7}{25} - \frac{i}{25}$

Evaluate (1+i)/(3+4i)

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