Simplify (3-2i)/(4-3i)

$\frac{3 - 2 i}{4 - 3 i}$

Step 1

Multiply the numerator and denominator of $\frac{3 - 2 i}{4 - 3 i}$ by the conjugate of $4 - 3 i$ to make the denominator real.

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

Step 2

Multiply.

Combine.

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

Simplify the numerator.

Expand $(3 - 2 i) (4 + 3 i)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $4$ .

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

Multiply $3$ by $3$ .

$\frac{12 + 9 i - 2 i \cdot 4 - 2 i (3 i)}{(4 - 3 i) (4 + 3 i)}$

Multiply $4$ by $- 2$ .

$\frac{12 + 9 i - 8 i - 2 i (3 i)}{(4 - 3 i) (4 + 3 i)}$

Multiply $- 2 i (3 i)$ .

Multiply $3$ by $- 2$ .

$\frac{12 + 9 i - 8 i - 6 i i}{(4 - 3 i) (4 + 3 i)}$

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

$\frac{12 + 9 i - 8 i - 6 (i^{1} i)}{(4 - 3 i) (4 + 3 i)}$

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

$\frac{12 + 9 i - 8 i - 6 (i^{1} i^{1})}{(4 - 3 i) (4 + 3 i)}$

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

$\frac{12 + 9 i - 8 i - 6 i^{1 + 1}}{(4 - 3 i) (4 + 3 i)}$

Add $1$ and $1$ .

$\frac{12 + 9 i - 8 i - 6 i^{2}}{(4 - 3 i) (4 + 3 i)}$

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

$\frac{12 + 9 i - 8 i - 6 \cdot - 1}{(4 - 3 i) (4 + 3 i)}$

Multiply $- 6$ by $- 1$ .

$\frac{12 + 9 i - 8 i + 6}{(4 - 3 i) (4 + 3 i)}$

Add $12$ and $6$ .

$\frac{18 + 9 i - 8 i}{(4 - 3 i) (4 + 3 i)}$

Subtract $8 i$ from $9 i$ .

$\frac{18 + i}{(4 - 3 i) (4 + 3 i)}$

Simplify the denominator.

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

Apply the distributive property.

$\frac{18 + i}{4 (4 + 3 i) - 3 i (4 + 3 i)}$

Apply the distributive property.

$\frac{18 + i}{4 \cdot 4 + 4 (3 i) - 3 i (4 + 3 i)}$

Apply the distributive property.

$\frac{18 + i}{4 \cdot 4 + 4 (3 i) - 3 i \cdot 4 - 3 i (3 i)}$

Simplify.

Multiply $4$ by $4$ .

$\frac{18 + i}{16 + 4 (3 i) - 3 i \cdot 4 - 3 i (3 i)}$

Multiply $3$ by $4$ .

$\frac{18 + i}{16 + 12 i - 3 i \cdot 4 - 3 i (3 i)}$

Multiply $4$ by $- 3$ .

$\frac{18 + i}{16 + 12 i - 12 i - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$\frac{18 + i}{16 + 12 i - 12 i - 9 i i}$

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

$\frac{18 + i}{16 + 12 i - 12 i - 9 (i^{1} i)}$

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

$\frac{18 + i}{16 + 12 i - 12 i - 9 (i^{1} i^{1})}$

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

$\frac{18 + i}{16 + 12 i - 12 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{18 + i}{16 + 12 i - 12 i - 9 i^{2}}$

Subtract $12 i$ from $12 i$ .

$\frac{18 + i}{16 + 0 - 9 i^{2}}$

Add $16$ and $0$ .

$\frac{18 + i}{16 - 9 i^{2}}$

Simplify each term.

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

$\frac{18 + i}{16 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{18 + i}{16 + 9}$

Add $16$ and $9$ .

$\frac{18 + i}{25}$

Step 3

Split the fraction $\frac{18 + i}{25}$ into two fractions.

$\frac{18}{25} + \frac{i}{25}$

Simplify (3-2i)/(4-3i)

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