Powers and Roots of Complex Numbers

What is (1/2 + i √3/2)^5?

1/2 - i √3/2

Consider the number z = 5 - 5i √3. What happens when z is raised to successively increasing powers? Use the drop-down menus to complete each sentence. The modulus increases by a factor of The argument increases by π over 3.

10 5

What is 5[cos(π/4) + i sin (π/4)] raised to the 3rd power?

125[cos(3π/4)+ i sin (3π/4)]

Consider the complex number z=27i. What are the characteristics of the cube root of z? Use the drop-down menus to complete each sentence. The cube roots are located on a circle with center at the pole and radius of The cube roots have arguments which differ by π over 3.

3 2

Determine which regions contain cube roots of -1. Check all that apply.

A.) on real axis C.) quadrant 1 F.) quadrant 4

Consider the graph of z^3. Which lettered point represents z?

B

Which of the following are square roots of -8+8i √3? Check all that apply.

B.) -2 -2i √3 C.) 2 + 2i √3

Determine which regions contain cube roots of 8i. Check all that apply.

B.) on imaginary axis C.) quadrant 1 D.) quadrant 2

Consider the graph of z. Which lettered point represents z²?

C

What is the correct justification for the indicated steps? Step C: Step D: Step E:

Step C: distributive property Step D: trigonometric sum identity Step E: factoring