Write in Standard Form (9-19i)/(1-4i)

$\frac{9 - 19 i}{1 - 4 i}$

Step 1

Multiply the numerator and denominator of $\frac{9 - 19 i}{1 - 4 i}$ by the conjugate of $1 - 4 i$ to make the denominator real.

$\frac{9 - 19 i}{1 - 4 i} \cdot \frac{1 + 4 i}{1 + 4 i}$

Step 2

Multiply.

Combine.

$\frac{(9 - 19 i) (1 + 4 i)}{(1 - 4 i) (1 + 4 i)}$

Simplify the numerator.

Expand $(9 - 19 i) (1 + 4 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{9 (1 + 4 i) - 19 i (1 + 4 i)}{(1 - 4 i) (1 + 4 i)}$

Apply the distributive property.

$\frac{9 \cdot 1 + 9 (4 i) - 19 i (1 + 4 i)}{(1 - 4 i) (1 + 4 i)}$

Apply the distributive property.

$\frac{9 \cdot 1 + 9 (4 i) - 19 i \cdot 1 - 19 i (4 i)}{(1 - 4 i) (1 + 4 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $9$ by $1$ .

$\frac{9 + 9 (4 i) - 19 i \cdot 1 - 19 i (4 i)}{(1 - 4 i) (1 + 4 i)}$

Multiply $4$ by $9$ .

$\frac{9 + 36 i - 19 i \cdot 1 - 19 i (4 i)}{(1 - 4 i) (1 + 4 i)}$

Multiply $- 19$ by $1$ .

$\frac{9 + 36 i - 19 i - 19 i (4 i)}{(1 - 4 i) (1 + 4 i)}$

Multiply $- 19 i (4 i)$ .

Multiply $4$ by $- 19$ .

$\frac{9 + 36 i - 19 i - 76 i i}{(1 - 4 i) (1 + 4 i)}$

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

$\frac{9 + 36 i - 19 i - 76 (i^{1} i)}{(1 - 4 i) (1 + 4 i)}$

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

$\frac{9 + 36 i - 19 i - 76 (i^{1} i^{1})}{(1 - 4 i) (1 + 4 i)}$

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

$\frac{9 + 36 i - 19 i - 76 i^{1 + 1}}{(1 - 4 i) (1 + 4 i)}$

Add $1$ and $1$ .

$\frac{9 + 36 i - 19 i - 76 i^{2}}{(1 - 4 i) (1 + 4 i)}$

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

$\frac{9 + 36 i - 19 i - 76 \cdot - 1}{(1 - 4 i) (1 + 4 i)}$

Multiply $- 76$ by $- 1$ .

$\frac{9 + 36 i - 19 i + 76}{(1 - 4 i) (1 + 4 i)}$

Add $9$ and $76$ .

$\frac{85 + 36 i - 19 i}{(1 - 4 i) (1 + 4 i)}$

Subtract $19 i$ from $36 i$ .

$\frac{85 + 17 i}{(1 - 4 i) (1 + 4 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{85 + 17 i}{1 (1 + 4 i) - 4 i (1 + 4 i)}$

Apply the distributive property.

$\frac{85 + 17 i}{1 \cdot 1 + 1 (4 i) - 4 i (1 + 4 i)}$

Apply the distributive property.

$\frac{85 + 17 i}{1 \cdot 1 + 1 (4 i) - 4 i \cdot 1 - 4 i (4 i)}$

Simplify.

Multiply $1$ by $1$ .

$\frac{85 + 17 i}{1 + 1 (4 i) - 4 i \cdot 1 - 4 i (4 i)}$

Multiply $4$ by $1$ .

$\frac{85 + 17 i}{1 + 4 i - 4 i \cdot 1 - 4 i (4 i)}$

Multiply $- 4$ by $1$ .

$\frac{85 + 17 i}{1 + 4 i - 4 i - 4 i (4 i)}$

Multiply $4$ by $- 4$ .

$\frac{85 + 17 i}{1 + 4 i - 4 i - 16 i i}$

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

$\frac{85 + 17 i}{1 + 4 i - 4 i - 16 (i^{1} i)}$

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

$\frac{85 + 17 i}{1 + 4 i - 4 i - 16 (i^{1} i^{1})}$

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

$\frac{85 + 17 i}{1 + 4 i - 4 i - 16 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{85 + 17 i}{1 + 4 i - 4 i - 16 i^{2}}$

Subtract $4 i$ from $4 i$ .

$\frac{85 + 17 i}{1 + 0 - 16 i^{2}}$

Add $1$ and $0$ .

$\frac{85 + 17 i}{1 - 16 i^{2}}$

Simplify each term.

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

$\frac{85 + 17 i}{1 - 16 \cdot - 1}$

Multiply $- 16$ by $- 1$ .

$\frac{85 + 17 i}{1 + 16}$

Add $1$ and $16$ .

$\frac{85 + 17 i}{17}$

Step 3

Cancel the common factor of $85 + 17 i$ and $17$ .

Factor $17$ out of $85$ .

$\frac{17 \cdot 5 + 17 i}{17}$

Factor $17$ out of $17 i$ .

$\frac{17 \cdot 5 + 17 (i)}{17}$

Factor $17$ out of $17 (5) + 17 (i)$ .

$\frac{17 (5 + i)}{17}$

Cancel the common factors.

Factor $17$ out of $17$ .

$\frac{17 (5 + i)}{17 (1)}$

Cancel the common factor.

$\frac{17 (5 + i)}{17 \cdot 1}$

Rewrite the expression.

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

Divide $5 + i$ by $1$ .

$5 + i$

Write in Standard Form (9-19i)/(1-4i)

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