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)