Write in Standard Form (5-i)/(4+3i)

$\frac{5 - i}{4 + 3 i}$

Step 1

Multiply the numerator and denominator of $\frac{5 - i}{4 + 3 i}$ by the conjugate of $4 + 3 i$ to make the denominator real.

$\frac{5 - i}{4 + 3 i} \cdot \frac{4 - 3 i}{4 - 3 i}$

Step 2

Multiply.

Combine.

$\frac{(5 - i) (4 - 3 i)}{(4 + 3 i) (4 - 3 i)}$

Simplify the numerator.

Expand $(5 - i) (4 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 (4 - 3 i) - i (4 - 3 i)}{(4 + 3 i) (4 - 3 i)}$

Apply the distributive property.

$\frac{5 \cdot 4 + 5 (- 3 i) - i (4 - 3 i)}{(4 + 3 i) (4 - 3 i)}$

Apply the distributive property.

$\frac{5 \cdot 4 + 5 (- 3 i) - i \cdot 4 - i (- 3 i)}{(4 + 3 i) (4 - 3 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $4$ .

$\frac{20 + 5 (- 3 i) - i \cdot 4 - i (- 3 i)}{(4 + 3 i) (4 - 3 i)}$

Multiply $- 3$ by $5$ .

$\frac{20 - 15 i - i \cdot 4 - i (- 3 i)}{(4 + 3 i) (4 - 3 i)}$

Multiply $4$ by $- 1$ .

$\frac{20 - 15 i - 4 i - i (- 3 i)}{(4 + 3 i) (4 - 3 i)}$

Multiply $- i (- 3 i)$ .

Multiply $- 3$ by $- 1$ .

$\frac{20 - 15 i - 4 i + 3 i i}{(4 + 3 i) (4 - 3 i)}$

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

$\frac{20 - 15 i - 4 i + 3 (i^{1} i)}{(4 + 3 i) (4 - 3 i)}$

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

$\frac{20 - 15 i - 4 i + 3 (i^{1} i^{1})}{(4 + 3 i) (4 - 3 i)}$

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

$\frac{20 - 15 i - 4 i + 3 i^{1 + 1}}{(4 + 3 i) (4 - 3 i)}$

Add $1$ and $1$ .

$\frac{20 - 15 i - 4 i + 3 i^{2}}{(4 + 3 i) (4 - 3 i)}$

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

$\frac{20 - 15 i - 4 i + 3 \cdot - 1}{(4 + 3 i) (4 - 3 i)}$

Multiply $3$ by $- 1$ .

$\frac{20 - 15 i - 4 i - 3}{(4 + 3 i) (4 - 3 i)}$

Subtract $3$ from $20$ .

$\frac{17 - 15 i - 4 i}{(4 + 3 i) (4 - 3 i)}$

Subtract $4 i$ from $- 15 i$ .

$\frac{17 - 19 i}{(4 + 3 i) (4 - 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{17 - 19 i}{4 (4 - 3 i) + 3 i (4 - 3 i)}$

Apply the distributive property.

$\frac{17 - 19 i}{4 \cdot 4 + 4 (- 3 i) + 3 i (4 - 3 i)}$

Apply the distributive property.

$\frac{17 - 19 i}{4 \cdot 4 + 4 (- 3 i) + 3 i \cdot 4 + 3 i (- 3 i)}$

Simplify.

Multiply $4$ by $4$ .

$\frac{17 - 19 i}{16 + 4 (- 3 i) + 3 i \cdot 4 + 3 i (- 3 i)}$

Multiply $- 3$ by $4$ .

$\frac{17 - 19 i}{16 - 12 i + 3 i \cdot 4 + 3 i (- 3 i)}$

Multiply $4$ by $3$ .

$\frac{17 - 19 i}{16 - 12 i + 12 i + 3 i (- 3 i)}$

Multiply $- 3$ by $3$ .

$\frac{17 - 19 i}{16 - 12 i + 12 i - 9 i i}$

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

$\frac{17 - 19 i}{16 - 12 i + 12 i - 9 (i^{1} i)}$

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

$\frac{17 - 19 i}{16 - 12 i + 12 i - 9 (i^{1} i^{1})}$

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

$\frac{17 - 19 i}{16 - 12 i + 12 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{17 - 19 i}{16 - 12 i + 12 i - 9 i^{2}}$

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

$\frac{17 - 19 i}{16 + 0 - 9 i^{2}}$

Add $16$ and $0$ .

$\frac{17 - 19 i}{16 - 9 i^{2}}$

Simplify each term.

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

$\frac{17 - 19 i}{16 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{17 - 19 i}{16 + 9}$

Add $16$ and $9$ .

$\frac{17 - 19 i}{25}$

Step 3

Split the fraction $\frac{17 - 19 i}{25}$ into two fractions.

$\frac{17}{25} + \frac{- 19 i}{25}$

Step 4

Move the negative in front of the fraction.

$\frac{17}{25} - \frac{19 i}{25}$

Write in Standard Form (5-i)/(4+3i)

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