Question

write the expression in standard form.

\\\frac{8 + 4i}{1 - i}\\

\\\frac{8 + 4i}{1 - i} = \box\\

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

Explanation:

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

Step 1: Identify the conjugate of the denominator

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

The conjugate of \( a - bi \) is \( a + bi \). For our denominator \( 1 - i \), the conjugate is:

$$ 1 + i $$

Step 2: Multiply the numerator and denominator by the conjugate

Set up the multiplication:

$$ \frac{8 + 4i}{1 - i} \cdot \frac{1 + i}{1 + i} = \frac{(8 + 4i)(1 + i)}{(1 - i)(1 + i)} $$

Step 3: Expand the numerator

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

$$ (8 + 4i)(1 + i) = 8(1) + 8(i) + 4i(1) + 4i(i) $$

$$ = 8 + 8i + 4i + 4i^2 $$

Since \( i^2 = -1 \), substitute \( -1 \) for \( i^2 \):

$$ = 8 + 12i + 4(-1) $$

$$ = 8 + 12i - 4 $$

$$ = 4 + 12i $$

Step 4: Expand the denominator

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

$$ (1 - i)(1 + i) = 1^2 - i^2 $$

$$ = 1 - (-1) $$

$$ = 1 + 1 $$

$$ = 2 $$

Step 5: Simplify the fraction

Put the simplified numerator and denominator back together:

$$ \frac{4 + 12i}{2} $$

Divide each term in the numerator by \( 2 \):

$$ \frac{4}{2} + \frac{12i}{2} = 2 + 6i $$

Answer:

$$ 2 + 6i $$