5.12. Complex Powers and Roots

Try these first

1. 1.

   What is the expression for z^2 if z = 3e^(iπ/6)?

   Hint

   Remember to square the modulus.

2. 2.

   Find the modulus of the complex number z = 1 + i.

   Hint

   Use Pythagorean theorem!

3. 3.

   What is the formula for complex power?

   • z^n = r^n e^(inθ)
   • z^n = r^n e^(θ)
   • z^n = re^(nθ)

   Hint

   Consider how the power affects both magnitude and direction.

4. 4.

   True or False: The nth roots of a complex number are always positioned randomly.

   • True
   • False

   Hint

   Think about symmetry in their distribution.

5. 5.

   Find all sixth roots of the complex number z = 64e^(iπ).

   Hint

   Remember to divide the angle by 6 and account for the full rotation with 2π.

6. 6.

   Solve for z^n = (1 + i) where n = 3 using exponentials.

   Hint

   Expressing initially in polar form simplifies the process.