Question

perform the indicated operation and simplify. write the answer in

\\\frac{7 + 3i}{2 - 3i}\\

\\\frac{7 + 3i}{2 - 3i} = \square\\

(simplify your answer. use integers or fractions for any numbers in the expression.)

Explanation:

🆕 New Concept Discovered: Division of Complex Numbers
Using conjugates to remove imaginary units from denominators.

Step 1: Multiply by the conjugate

To divide complex numbers, multiply the numerator and the denominator by the complex conjugate of the denominator. The conjugate of \(2 - 3i\) is \(2 + 3i\).

$$ \frac{7 + 3i}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i} = \frac{(7 + 3i)(2 + 3i)}{(2 - 3i)(2 + 3i)} $$

Step 2: Expand the numerator and denominator

Multiply the terms in the numerator using the FOIL method:

$$ (7 + 3i)(2 + 3i) = 14 + 21i + 6i + 9i^2 $$

Multiply the terms in the denominator (difference of squares):

$$ (2 - 3i)(2 + 3i) = 2^2 - (3i)^2 = 4 - 9i^2 $$

Step 3: Simplify using \(i^2 = -1\)

Substitute \(i^2 = -1\) into both expressions:

Numerator:

$$ 14 + 27i + 9(-1) = 14 + 27i - 9 = 5 + 27i $$

Denominator:

$$ 4 - 9(-1) = 4 + 9 = 13 $$

Step 4: Write in standard form \(a + bi\)

Combine the simplified numerator and denominator:

$$ \frac{5 + 27i}{13} = \frac{5}{13} + \frac{27}{13}i $$

Answer:

$$ \frac{5}{13} + \frac{27}{13}i $$