Write in Standard Form (1-2i)/(2-3i)

$\frac{(1 - 2 i)}{(2 - 3 i)}$

Step 1

Multiply the numerator and denominator of $\frac{1 - 2 i}{2 - 3 i}$ by the conjugate of $2 - 3 i$ to make the denominator real.

$\frac{1 - 2 i}{2 - 3 i} \cdot \frac{2 + 3 i}{2 + 3 i}$

Step 2

Multiply.

Combine.

$\frac{(1 - 2 i) (2 + 3 i)}{(2 - 3 i) (2 + 3 i)}$

Simplify the numerator.

Expand $(1 - 2 i) (2 + 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{1 (2 + 3 i) - 2 i (2 + 3 i)}{(2 - 3 i) (2 + 3 i)}$

Apply the distributive property.

$\frac{1 \cdot 2 + 1 (3 i) - 2 i (2 + 3 i)}{(2 - 3 i) (2 + 3 i)}$

Apply the distributive property.

$\frac{1 \cdot 2 + 1 (3 i) - 2 i \cdot 2 - 2 i (3 i)}{(2 - 3 i) (2 + 3 i)}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $1$ .

$\frac{2 + 1 (3 i) - 2 i \cdot 2 - 2 i (3 i)}{(2 - 3 i) (2 + 3 i)}$

Multiply $3 i$ by $1$ .

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

Multiply $2$ by $- 2$ .

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

Multiply $- 2 i (3 i)$ .

Multiply $3$ by $- 2$ .

$\frac{2 + 3 i - 4 i - 6 i i}{(2 - 3 i) (2 + 3 i)}$

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

$\frac{2 + 3 i - 4 i - 6 (i^{1} i)}{(2 - 3 i) (2 + 3 i)}$

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

$\frac{2 + 3 i - 4 i - 6 (i^{1} i^{1})}{(2 - 3 i) (2 + 3 i)}$

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

$\frac{2 + 3 i - 4 i - 6 i^{1 + 1}}{(2 - 3 i) (2 + 3 i)}$

Add $1$ and $1$ .

$\frac{2 + 3 i - 4 i - 6 i^{2}}{(2 - 3 i) (2 + 3 i)}$

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

$\frac{2 + 3 i - 4 i - 6 \cdot - 1}{(2 - 3 i) (2 + 3 i)}$

Multiply $- 6$ by $- 1$ .

$\frac{2 + 3 i - 4 i + 6}{(2 - 3 i) (2 + 3 i)}$

Add $2$ and $6$ .

$\frac{8 + 3 i - 4 i}{(2 - 3 i) (2 + 3 i)}$

Subtract $4 i$ from $3 i$ .

$\frac{8 - i}{(2 - 3 i) (2 + 3 i)}$

Simplify the denominator.

Expand $(2 - 3 i) (2 + 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{8 - i}{2 (2 + 3 i) - 3 i (2 + 3 i)}$

Apply the distributive property.

$\frac{8 - i}{2 \cdot 2 + 2 (3 i) - 3 i (2 + 3 i)}$

Apply the distributive property.

$\frac{8 - i}{2 \cdot 2 + 2 (3 i) - 3 i \cdot 2 - 3 i (3 i)}$

Simplify.

Multiply $2$ by $2$ .

$\frac{8 - i}{4 + 2 (3 i) - 3 i \cdot 2 - 3 i (3 i)}$

Multiply $3$ by $2$ .

$\frac{8 - i}{4 + 6 i - 3 i \cdot 2 - 3 i (3 i)}$

Multiply $2$ by $- 3$ .

$\frac{8 - i}{4 + 6 i - 6 i - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$\frac{8 - i}{4 + 6 i - 6 i - 9 i i}$

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

$\frac{8 - i}{4 + 6 i - 6 i - 9 (i^{1} i)}$

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

$\frac{8 - i}{4 + 6 i - 6 i - 9 (i^{1} i^{1})}$

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

$\frac{8 - i}{4 + 6 i - 6 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{8 - i}{4 + 6 i - 6 i - 9 i^{2}}$

Subtract $6 i$ from $6 i$ .

$\frac{8 - i}{4 + 0 - 9 i^{2}}$

Add $4$ and $0$ .

$\frac{8 - i}{4 - 9 i^{2}}$

Simplify each term.

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

$\frac{8 - i}{4 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{8 - i}{4 + 9}$

Add $4$ and $9$ .

$\frac{8 - i}{13}$

Step 3

Split the fraction $\frac{8 - i}{13}$ into two fractions.

$\frac{8}{13} + \frac{- i}{13}$

Step 4

Move the negative in front of the fraction.

$\frac{8}{13} - \frac{i}{13}$

Write in Standard Form (1-2i)/(2-3i)

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