To find the fifth roots of the complex number given in polar form as 32(cos(π/3) + isin(π/3)), we first identify the modulus and argument of the complex number.
The modulus of the complex number is 32, and the argument (angle) is π/3. To calculate the fifth roots, we use the formula for the nth roots of a complex number:
r^{1/n} (cos((θ + 2kπ)/n) + isin((θ + 2kπ)/n)) for k = 0, 1, 2, 3, …, n-1.
Here, r = 32 (the modulus) and θ = π/3 (the argument), with n = 5 (since we are looking for the fifth roots).
1. First, we compute the modulus of the fifth root:
r^{1/5} = 32^{1/5} = 2.
2. Next, we compute the angles for k = 0, 1, 2, 3, 4:
- For k = 0: θ_0 = (π/3 + 2(0)π) / 5 = π/15
- For k = 1: θ_1 = (π/3 + 2(1)π) / 5 = (π/15) + (2π/5) = 7π/15
- For k = 2: θ_2 = (π/3 + 2(2)π) / 5 = (π/15) + (4π/5) = 13π/15
- For k = 3: θ_3 = (π/3 + 2(3)π) / 5 = (π/15) + (6π/5) = 19π/15
- For k = 4: θ_4 = (π/3 + 2(4)π) / 5 = (π/15) + (8π/5) = 25π/15 = 5π/3
3. Finally, compiling all the fifth roots, we have:
- Root 1:
2(cos(π/15) + isin(π/15)) - Root 2:
2(cos(7π/15) + isin(7π/15)) - Root 3:
2(cos(13π/15) + isin(13π/15)) - Root 4:
2(cos(19π/15) + isin(19π/15)) - Root 5:
2(cos(5π/3) + isin(5π/3))
These roots can be expressed in rectangular form by calculating the cosine and sine values. Choose any of these to answer your question about the fifth root of the complex number.