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 i^{2} = -1.

## Formula to Calculate the Power of a Complex Number

The complex number power formula is given below.

z^{n} = (re^{Iθ })^{n} = r^{n} e^{inθ} |

**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)^{1/2}e^{(i67.38)/2}

√(5 + 12i) = √13 e^{i33.69}

= √13 (cos 33.69 + i sin 33.69)