Evaluate the expression (-4 - 1i)(1i) / (1 - 1i) and write the result in the form a + bi. Then a = ___ and...

Question

Evaluate the expression (-4 - 1i)(1i) / (1 - 1i) and write the result in the form a + bi. Then a = ___ and b = ___.

SnapAndSolve AI · Accepted Answer

First, simplify the numerator:
(-4 - 1i)(1i) = -4i - 1i^2 = -4i + 1 (since i^2 = -1).
So, the numerator is 1 - 4i.

Now, simplify the denominator:
1 - 1i.

To divide, multiply numerator and denominator by the conjugate of the denominator:
(1 - 4i)(1 + 1i) / ((1 - 1i)(1 + 1i)).

The denominator becomes:
(1 - 1i)(1 + 1i) = 1^2 - (1i)^2 = 1 - (-1) = 2.

The numerator becomes:
(1 - 4i)(1 + 1i) = 1 + 1i - 4i - 4i^2 = 1 - 3i + 4 = 5 - 3i.

So, the expression is:
(5 - 3i) / 2 = 5/2 - (3/2)i.

Thus, a = 5/2 and b = -3/2.

Answer