Simplify:
(4+2i)/2i(4-2i)^-1
To simplify the expression, let's break it down step by step.
First, let's simplify the denominator, which is (4 - 2i)^(-1).
Recall that the power of -1 in the denominator means we need to take the reciprocal of the term inside the parentheses. In this case, we need to find the reciprocal of (4 - 2i).
To find the reciprocal of a complex number, we multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of (4 - 2i) is (4 + 2i).
So, the reciprocal of (4 - 2i) is ((4 + 2i)/(4 + 2i)).
Now, let's multiply the numerator (4 + 2i) by the reciprocal of the denominator ((4 + 2i)/(4 + 2i)).
(4 + 2i) / [(2i) * ((4 + 2i)/(4 + 2i))]
Now, let's simplify the expression further.
First, let's multiply the numerator (4 + 2i) by the reciprocal of the denominator ((4 + 2i)/(4 + 2i)).
(4 + 2i) * (4 + 2i) / (2i)
To multiply the complex numbers in the numerator, we'll use the FOIL method.
(4 + 2i) * (4 + 2i) = 16 + 8i + 8i + 4i^2
Remember that i^2 is equal to -1.
So, the numerator simplifies to:
16 + 8i + 8i - 4
Combine the like terms:
12 + 16i - 4
8 + 16i
Combining all the steps, the simplified expression is:
(8 + 16i) / (2i)