Question

divide.

\\\frac{4}{9+i}\\

\\\frac{4}{9+i} = \square\\

(simplify your answer. type your answer in the

Explanation:

🆕 New Concept Discovered: Dividing Complex Numbers
Multiply by the conjugate to clear the imaginary unit 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 \(9 + i\) is \(9 - i\).

$$ \frac{4}{9 + i} \cdot \frac{9 - i}{9 - i} = \frac{4(9 - i)}{(9 + i)(9 - i)} $$

Step 2: Simplify the denominator

Using the difference of squares formula, \((a + b)(a - b) = a^2 - b^2\), and the fact that \(i^2 = -1\):

$$ (9 + i)(9 - i) = 9^2 - i^2 = 81 - (-1) = 81 + 1 = 82 $$

Step 3: Simplify the numerator and the fraction

Now, substitute the simplified denominator back into the expression:

$$ \frac{4(9 - i)}{82} = \frac{36 - 4i}{82} $$

Divide each term by the greatest common divisor, which is \(2\):

$$ \frac{36}{82} - \frac{4i}{82} = \frac{18}{41} - \frac{2}{41}i $$

Answer:

$$ \frac{18}{41} - \frac{2}{41}i $$