Write in Standard Form (-1+3i)/(4-i)

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

Step 1

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

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $4$ .

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

Rewrite $- 1 i$ as $- i$ .

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

Multiply $4$ by $3$ .

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

Multiply $3 i i$ .

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

$\frac{- 4 - i + 12 i + 3 (i^{1} i)}{(4 - i) (4 + i)}$

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

$\frac{- 4 - i + 12 i + 3 (i^{1} i^{1})}{(4 - i) (4 + i)}$

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

$\frac{- 4 - i + 12 i + 3 i^{1 + 1}}{(4 - i) (4 + i)}$

Add $1$ and $1$ .

$\frac{- 4 - i + 12 i + 3 i^{2}}{(4 - i) (4 + i)}$

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

$\frac{- 4 - i + 12 i + 3 \cdot - 1}{(4 - i) (4 + i)}$

Multiply $3$ by $- 1$ .

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

Subtract $3$ from $- 4$ .

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

Add $- i$ and $12 i$ .

$\frac{- 7 + 11 i}{(4 - i) (4 + i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{- 7 + 11 i}{4 (4 + i) - i (4 + i)}$

Apply the distributive property.

$\frac{- 7 + 11 i}{4 \cdot 4 + 4 i - i (4 + i)}$

Apply the distributive property.

$\frac{- 7 + 11 i}{4 \cdot 4 + 4 i - i \cdot 4 - i i}$

Simplify.

Multiply $4$ by $4$ .

$\frac{- 7 + 11 i}{16 + 4 i - i \cdot 4 - i i}$

Multiply $4$ by $- 1$ .

$\frac{- 7 + 11 i}{16 + 4 i - 4 i - i i}$

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

$\frac{- 7 + 11 i}{16 + 4 i - 4 i - (i^{1} i)}$

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

$\frac{- 7 + 11 i}{16 + 4 i - 4 i - (i^{1} i^{1})}$

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

$\frac{- 7 + 11 i}{16 + 4 i - 4 i - i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 7 + 11 i}{16 + 4 i - 4 i - i^{2}}$

Subtract $4 i$ from $4 i$ .

$\frac{- 7 + 11 i}{16 + 0 - i^{2}}$

Add $16$ and $0$ .

$\frac{- 7 + 11 i}{16 - i^{2}}$

Simplify each term.

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

$\frac{- 7 + 11 i}{16 - - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{- 7 + 11 i}{16 + 1}$

Add $16$ and $1$ .

$\frac{- 7 + 11 i}{17}$

Step 3

Split the fraction $\frac{- 7 + 11 i}{17}$ into two fractions.

$\frac{- 7}{17} + \frac{11 i}{17}$

Step 4

Move the negative in front of the fraction.

$- \frac{7}{17} + \frac{11 i}{17}$

Write in Standard Form (-1+3i)/(4-i)

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