Question

Evaluate the cube root of z = 27 cis(240 degrees). Then raise them to the cube. Show the steps of your reasoning.

Answer 1

To evaluate the cube root of z = 27 cis(240 degrees), we first find the magnitude of the complex number, then divide the angle by 3. The cube root of z is 3 cis(80 degrees), and the cube of the cube root is 27 cis(240 degrees).

Step-by-step explanation:

To evaluate the cube root of z = 27 cis(240 degrees), we can use the fact that the cube root of a complex number can be found by taking the cube root of its magnitude and dividing the angle by 3.

1. First, we find the magnitude of 27 cis(240 degrees), which is the cube root of 27, which is 3.
2. Next, we divide the angle 240 degrees by 3, giving us 80 degrees.
3. Therefore, the cube root of z = 27 cis(240 degrees) is 3 cis(80 degrees).
4. If we raise 3 cis(80 degrees) to the cube, we cube the magnitude (3^3 = 27) and multiply the angle by 3 (80 degrees * 3 = 240 degrees).
5. So, the cube of the cube root of z = 27 cis(240 degrees) is 27 cis(240 degrees).