How Do You Find the nth Power of a Complex Number?

The easiest way to find the nth power of a complex number is to convert it to its polar (or exponential) form and then apply DeMoivre's Theorem.

Steps to Calculate the nth Power of a Complex Number

1. Convert to Polar Form: Express the complex number z = a + bi in polar form, z = r(cos θ + i sin θ) or z = r e^(iθ). Here, r is the modulus (or absolute value) of z, calculated as r = √(a² + b²), and θ is the argument of z, found using θ = arctan(b/a), considering the quadrant of z.

2. Apply DeMoivre's Theorem: DeMoivre's Theorem states that for any complex number in polar form z = r(cos θ + i sin θ) and any integer n:

   • zⁿ = [r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ))
   • In exponential form: zⁿ = (r e^(iθ))ⁿ = rⁿ e^(inθ)

3. Convert Back (Optional): If desired, you can convert the result back to rectangular form a + bi by calculating cos(nθ) and sin(nθ) and distributing rⁿ:

   • zⁿ = rⁿ cos(nθ) + i rⁿ sin(nθ)

Example

Let's find (1 + i)⁴:

1. Convert to Polar Form:

   • r = √(1² + 1²) = √2
   • θ = arctan(1/1) = π/4
   • So, 1 + i = √2 (cos(π/4) + i sin(π/4)) = √2 e^(iπ/4)

2. Apply DeMoivre's Theorem:

   • (1 + i)⁴ = (√2)⁴ (cos(4π/4) + i sin(4π/4)) = 4(cos(π) + i sin(π))
   • Or: (1 + i)⁴ = (√2 e^(iπ/4))⁴ = 4e^(iπ)

3. Convert Back to Rectangular Form:

   • 4(cos(π) + i sin(π)) = 4(-1 + i 0) = -4*

Therefore, (1 + i)⁴ = -4.

Summary

Finding the nth power of a complex number is simplified by converting it to polar or exponential form and applying DeMoivre's Theorem. This approach avoids complex multiplication and makes the calculation more manageable, especially for large values of n.