Question

perform the indicated operation and simplify. write the answer in the form \\(a + bi\\).

\\\frac{6 - 3i}{7 + 2i}\\

\\\frac{6 - 3i}{7 + 2i} = \square\\

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

Explanation:

🆕 New Concept Discovered: Dividing Complex Numbers
Using the complex conjugate to clear imaginary units from the denominator.

Step 1: Multiply by the conjugate

To divide complex numbers, we multiply both the numerator and the denominator by the complex conjugate of the denominator. The conjugate of \(7 + 2i\) is \(7 - 2i\).

$$ \frac{6 - 3i}{7 + 2i} \cdot \frac{7 - 2i}{7 - 2i} $$

Step 2: Expand the numerator

Multiply the terms in the numerator using the FOIL method (First, Outer, Inner, Last):

$$ (6 - 3i)(7 - 2i) = 6(7) + 6(-2i) - 3i(7) - 3i(-2i) $$

$$ = 42 - 12i - 21i + 6i^2 $$

Since \(i^2 = -1\):

$$ = 42 - 33i + 6(-1) $$

$$ = 42 - 33i - 6 $$

$$ = 36 - 33i $$

Step 3: Expand the denominator

Multiply the terms in the denominator. Since they are conjugates, the middle terms cancel out:

$$ (7 + 2i)(7 - 2i) = 7^2 - (2i)^2 $$

$$ = 49 - 4i^2 $$

Since \(i^2 = -1\):

$$ = 49 - 4(-1) $$

$$ = 49 + 4 $$

$$ = 53 $$

Step 4: Simplify into standard form

Combine the simplified numerator and denominator, then write the result in the standard form \(a + bi\):

$$ \frac{36 - 33i}{53} = \frac{36}{53} - \frac{33}{53}i $$

Answer:

$$ \frac{36}{53} - \frac{33}{53}i $$