Question

question 5

\frac{8-5i}{2-4i}

\bigcirc \frac{2}{5} + \frac{4}{5}i
\bigcirc \frac{1}{5} + \frac{2}{5}i
\bigcirc \frac{60}{41} + \frac{7}{41}i
\bigcirc \frac{9}{5} + \frac{11}{10}i

Explanation:

🆕 New Concept Discovered: Dividing Complex Numbers
Using the conjugate to remove imaginary parts 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 \(2 - 4i\) is \(2 + 4i\).

$$ \frac{8 - 5i}{2 - 4i} \cdot \frac{2 + 4i}{2 + 4i} = \frac{(8 - 5i)(2 + 4i)}{(2 - 4i)(2 + 4i)} $$

Step 2: Expand the numerator

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

$$ (8 - 5i)(2 + 4i) = 8(2) + 8(4i) - 5i(2) - 5i(4i) $$

$$ = 16 + 32i - 10i - 20i^2 $$

Since \(i^2 = -1\), we substitute and simplify:

$$ = 16 + 22i - 20(-1) $$

$$ = 16 + 22i + 20 $$

$$ = 36 + 22i $$

Step 3: Expand the denominator

Multiply the terms in the denominator:

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

$$ = 4 - 16i^2 $$

$$ = 4 - 16(-1) $$

$$ = 4 + 16 $$

$$ = 20 $$

Step 4: Simplify the fraction

Combine the simplified numerator and denominator:

$$ \frac{36 + 22i}{20} = \frac{36}{20} + \frac{22}{20}i $$

Reduce each fraction to its simplest form by dividing the numerators and denominators by their greatest common divisors:

$$ \frac{36}{20} = \frac{9}{5} $$

$$ \frac{22}{20} = \frac{11}{10} $$

This gives:

$$ \frac{9}{5} + \frac{11}{10}i $$

Answer:

$$ \frac{9}{5} + \frac{11}{10}i $$