Question

perform the indicated operation.

\\\frac{-5 + 4i}{5i}\\

\\\frac{-5 + 4i}{5i} = \square\\

(simplify your answer. type your answer in

Explanation:

🆕 New Concept Discovered: Dividing Complex Numbers
Removing imaginary units from the denominator.

Step 1: Multiply numerator and denominator by \( i \)

To eliminate the imaginary unit \( i \) from the denominator, we multiply both the top and the bottom of the fraction by \( i \). This is similar to rationalizing a denominator.

$$ \frac{-5 + 4i}{5i} \cdot \frac{i}{i} = \frac{(-5 + 4i)i}{5i^2} $$

Step 2: Distribute and simplify

Distribute \( i \) in the numerator and use the definition \( i^2 = -1 \) to simplify both the numerator and the denominator.

$$ \frac{-5i + 4i^2}{5(-1)} $$

$$ \frac{-5i + 4(-1)}{-5} $$

$$ \frac{-5i - 4}{-5} $$

Step 3: Write in standard form \( a + bi \)

Separate the fraction into its real and imaginary parts and simplify the signs.

$$ \frac{-4}{-5} + \frac{-5i}{-5} = \frac{4}{5} + i $$

Answer:

$$ \frac{4}{5} + i $$