Divide (6-i)/(5+4i)

$\frac{(6 - i)}{(5 + 4 i)}$

Multiply the numerator and denominator of $\frac{6 - i}{5 + 4 i}$ by the conjugate of $5 + 4 i$ to make the denominator real.

$\frac{6 - i}{5 + 4 i} \cdot \frac{5 - 4 i}{5 - 4 i}$

Multiply.

Combine.

$\frac{(6 - i) (5 - 4 i)}{(5 + 4 i) (5 - 4 i)}$

Simplify the numerator.

Expand $(6 - i) (5 - 4 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{6 (5 - 4 i) - i (5 - 4 i)}{(5 + 4 i) (5 - 4 i)}$

Apply the distributive property.

$\frac{6 \cdot 5 + 6 (- 4 i) - i (5 - 4 i)}{(5 + 4 i) (5 - 4 i)}$

Apply the distributive property.

$\frac{6 \cdot 5 + 6 (- 4 i) - i \cdot 5 - i (- 4 i)}{(5 + 4 i) (5 - 4 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $5$ .

$\frac{30 + 6 (- 4 i) - i \cdot 5 - i (- 4 i)}{(5 + 4 i) (5 - 4 i)}$

Multiply $- 4$ by $6$ .

$\frac{30 - 24 i - i \cdot 5 - i (- 4 i)}{(5 + 4 i) (5 - 4 i)}$

Multiply $5$ by $- 1$ .

$\frac{30 - 24 i - 5 i - i (- 4 i)}{(5 + 4 i) (5 - 4 i)}$

Multiply $- i (- 4 i)$ .

Multiply $- 4$ by $- 1$ .

$\frac{30 - 24 i - 5 i + 4 i i}{(5 + 4 i) (5 - 4 i)}$

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

$\frac{30 - 24 i - 5 i + 4 (i^{1} i)}{(5 + 4 i) (5 - 4 i)}$

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

$\frac{30 - 24 i - 5 i + 4 (i^{1} i^{1})}{(5 + 4 i) (5 - 4 i)}$

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

$\frac{30 - 24 i - 5 i + 4 i^{1 + 1}}{(5 + 4 i) (5 - 4 i)}$

Add $1$ and $1$ .

$\frac{30 - 24 i - 5 i + 4 i^{2}}{(5 + 4 i) (5 - 4 i)}$

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

$\frac{30 - 24 i - 5 i + 4 \cdot - 1}{(5 + 4 i) (5 - 4 i)}$

Multiply $4$ by $- 1$ .

$\frac{30 - 24 i - 5 i - 4}{(5 + 4 i) (5 - 4 i)}$

Subtract $4$ from $30$ .

$\frac{26 - 24 i - 5 i}{(5 + 4 i) (5 - 4 i)}$

Subtract $5 i$ from $- 24 i$ .

$\frac{26 - 29 i}{(5 + 4 i) (5 - 4 i)}$

Simplify the denominator.

Expand $(5 + 4 i) (5 - 4 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{26 - 29 i}{5 (5 - 4 i) + 4 i (5 - 4 i)}$

Apply the distributive property.

$\frac{26 - 29 i}{5 \cdot 5 + 5 (- 4 i) + 4 i (5 - 4 i)}$

Apply the distributive property.

$\frac{26 - 29 i}{5 \cdot 5 + 5 (- 4 i) + 4 i \cdot 5 + 4 i (- 4 i)}$

Simplify.

Multiply $5$ by $5$ .

$\frac{26 - 29 i}{25 + 5 (- 4 i) + 4 i \cdot 5 + 4 i (- 4 i)}$

Multiply $- 4$ by $5$ .

$\frac{26 - 29 i}{25 - 20 i + 4 i \cdot 5 + 4 i (- 4 i)}$

Multiply $5$ by $4$ .

$\frac{26 - 29 i}{25 - 20 i + 20 i + 4 i (- 4 i)}$

Multiply $- 4$ by $4$ .

$\frac{26 - 29 i}{25 - 20 i + 20 i - 16 i i}$

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

$\frac{26 - 29 i}{25 - 20 i + 20 i - 16 (i^{1} i)}$

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

$\frac{26 - 29 i}{25 - 20 i + 20 i - 16 (i^{1} i^{1})}$

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

$\frac{26 - 29 i}{25 - 20 i + 20 i - 16 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{26 - 29 i}{25 - 20 i + 20 i - 16 i^{2}}$

Add $- 20 i$ and $20 i$ .

$\frac{26 - 29 i}{25 + 0 - 16 i^{2}}$

Add $25$ and $0$ .

$\frac{26 - 29 i}{25 - 16 i^{2}}$

Simplify each term.

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

$\frac{26 - 29 i}{25 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$\frac{26 - 29 i}{25 + 16}$

Add $25$ and $16$ .

$\frac{26 - 29 i}{41}$

Split the fraction $\frac{26 - 29 i}{41}$ into two fractions.

$\frac{26}{41} + \frac{- 29 i}{41}$

Move the negative in front of the fraction.

$\frac{26}{41} - \frac{29 i}{41}$

Divide (6-i)/(5+4i)

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