Question

divide.

\\\frac{-2}{-7 - 9i}\\

\\\frac{-2}{-7 - 9i} = \square\\

(simplify your answer. type your answer in the form \\(a + bi\\).)

Explanation:

🆕 New Concept Discovered: Dividing Complex Numbers
Multiply numerator and denominator by the complex conjugate.

Step 1: Identify the complex conjugate

To divide by a complex number, we need to eliminate the imaginary unit \( i \) from the denominator. We do this by multiplying both the top and the bottom of the fraction by the complex conjugate of the denominator.

The denominator is \( -7 - 9i \).

Its complex conjugate is found by changing the sign of the imaginary part:

$$ -7 + 9i $$

Step 2: Multiply numerator and denominator

Now, multiply both the numerator and the denominator by this conjugate:

$$ \frac{-2}{-7 - 9i} \cdot \frac{-7 + 9i}{-7 + 9i} $$

Distribute the numerator:

$$ -2(-7 + 14i) = 14 - 18i $$

Step 3: Simplify the denominator

Multiply the conjugate pair in the denominator using the difference of squares formula, \( (a-bi)(a+bi) = a^2 + b^2 \):

$$ (-7 - 9i)(-7 + 9i) = (-7)^2 - (9i)^2 $$

$$ = 49 - 81i^2 $$

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

$$ = 49 - 81(-1) $$

$$ = 49 + 81 $$

$$ = 130 $$

Step 4: Write in standard form

Combine the simplified numerator and denominator:

$$ \frac{14 - 18i}{130} $$

Split the fraction to write the complex number in standard form \( a + bi \):

$$ \frac{14}{130} - \frac{18}{130}i $$

Reduce both fractions by dividing the numerators and denominators by their greatest common divisor, which is \( 2 \):

$$ \frac{7}{65} - \frac{9}{65}i $$

Answer:

$$ \frac{7}{65} - \frac{9}{65}i $$