# 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,Â 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.

 zn = (reIÎ¸ )n = rn einÎ¸

Example Question

Question 1:Compute: (3+3i)5

Solution:

Here is the exponential form of 3+3i

r = âˆš (9+9) = 3 âˆš2

tan Î¸ = (3/3)

â‡’arg z = (Ï€/4)

3 + 3i = 3âˆš2ei(Ï€/4)

Now, (3 + 3i)5 = (3 âˆš2)5 ei(5Ï€/4)

= 972 âˆš2 (cos(5Ï€/4) + isin (5Ï€/4))

= 972 âˆš2 [(âˆ’âˆš2/2)âˆ’(âˆš2/2)i]

= âˆ’ 972 âˆ’ 972i

Question 2: Compute: (1 – âˆš3i)6
Solution:
Given complex number is (1 – âˆš3i)6
The exponential form of 1 – âˆš3i is:
r = âˆš(1+3) = 2
tan Î¸ = (âˆš3/1)
â‡’arg z = (Ï€/3)
1 – âˆš3i = 2ei(Ï€/3)
Now, (1 – âˆš3i)6 = (2)6ei(6Ï€/3)
= 64 eiÏ€
= 64 [cos Ï€ + i sin Ï€] = 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 = âˆš(5 + 12i) = (5 + 12i)Â½
r = âˆš(25 + 144) = âˆš169 = 13
tan Î¸ = (12/5)
Î¸ = tan-1(12/5) = 67.38
â‡’arg z = 67.38
5 + 12i = 13ei67.38
(5 + 12i)Â½ = (13)1/2e(i67.38)/2
âˆš(5 + 12i) = âˆš13 ei33.69
= âˆš13 (cos 33.69 + i sin 33.69)