Complex Number Power Formula

The complex number power formula is used to compute the value of a complex number which is raised to the power of “n”. To recall, a complex number is the form of x + iy, where x and y are the real numbers and “i” is an imaginary number. The “i” satisfies i² = -1.

Formula to Calculate the Power of a Complex Number

The complex number power formula is given below.

Example Question

Question 1:Compute: (3+3i)⁵

Solution:

Here is the exponential form of 3+3i

r = √ (9+9) = 3 √2

tan θ = (3/3)

⇒arg z = (π/4)

3 + 3i = 3√2e^i(π/4)

Now, (3 + 3i)⁵ = (3 √2)⁵ e^i(5π/4)

= 972 √2 (cos(5π/4) + isin (5π/4))

= 972 √2 [(−√2/2)−(√2/2)i]

= − 972 − 972i

Question 2: Compute: (1 – √3i)⁶
Solution:
Given complex number is (1 – √3i)⁶
The exponential form of 1 – √3i is:
r = √(1+3) = 2
tan θ = (√3/1)
⇒arg z = (π/3)
1 – √3i = 2e^i(π/3)
Now, (1 – √3i)⁶ = (2)6e^i(6π/3)
= 64 e^iπ
= 64 [cos π + i sin π] = 64[1 + i(0)] = 64

Question 3: Write the square root of 5 + 12i in the polar form.

Solution:
Given complex number is: 5 + 12i
Square root of the given complex number = √(5 + 12i) = (5 + 12i)^½
r = √(25 + 144) = √169 = 13
tan θ = (12/5)
θ = tan-1(12/5) = 67.38
⇒arg z = 67.38
5 + 12i = 13^ei67.38
(5 + 12i)^½ = (13)^1/2e^(i67.38)/2
√(5 + 12i) = √13 e^i33.69
= √13 (cos 33.69 + i sin 33.69)