i^59 and 4
____
3+5i
To simplify the expression (i^59) / (4 / (3+5i)), we can start by simplifying the numerator and denominator separately.
Let's begin with the numerator, i^59. The complex number i represents the imaginary unit, which is defined as the square root of -1. It follows a pattern: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This pattern repeats every four powers of i.
In this case, i^59 can be rewritten as (i^56 * i^3), where i^56 is equivalent to (i^4)^14, which simplifies to 1^14 = 1. Therefore, i^59 simplifies to (1 * i^3).
Next, let us simplify the denominator, which is (4 / (3+5i)). To divide a complex number by another complex number, we need to multiply both the numerator and denominator by the conjugate of the denominator, which is (3-5i).
Multiplying the numerator and denominator by (3-5i), we get:
[(4 * (3-5i)) / ((3+5i) * (3-5i))].
Now, let's simplify this expression further:
(4 * (3-5i)) represents a multiplication of a real number by a complex number, which results in a complex number.
((3+5i) * (3-5i)) represents a multiplication of two complex numbers, which results in a real number.
We can calculate the values of the real and complex parts step-by-step:
(4 * (3-5i)) = (12 - 20i)
((3+5i) * (3-5i)) = (9 - 15i + 15i - 25i^2) = (9 - 25(-1)) = (9 + 25) = 34
So, the simplified expression becomes: (12 - 20i) / 34.
In conclusion, the simplified expression (i^59) / (4 / (3+5i)) is equal to (12 - 20i) / 34.