Final answer:
To find the complex fifth roots of (5 - 5√3i), convert the number to polar form and use De Moivre's theorem, which gives distinct roots for each k value (0 to 4) using the formula involving the modulus and argument of the complex number.
Step-by-step explanation:
The question is about finding the complex fifth roots of the number (5 - 5√3i). To find the fifth roots of a complex number, one must first express the number in polar form and then apply De Moivre's theorem, which tells us that the nth roots of a complex number a+bi are given by:
r^(1/n) * [ cos(θ/n + 2kπ/n) + i*sin(θ/n + 2kπ/n) ]
for k = 0, 1, 2, ..., n-1 where r is the modulus of the complex number and θ (theta) is the argument. Since we are looking for fifth roots, n=5, and k will take on the values 0, 1, 2, 3, and 4. Each value of k provides us with a distinct fifth root.