Question

3. Solve the following equations:

a) z3 + 8i = 0
b) z5 − 1 − i = 0

Answer 1

[AI-generated answer — Bot GPT 3.5]

a) To solve for z, we need to find the cube root of -8i.

First, we convert -8i to polar form:
-8i = 8*(cos(270°) + isin(270°))

Now, z = (8)^(1/3)*(cos((270°)/3 + n*360°/3) + isin((270°)/3 + n*360°/3)), where n = 0, 1, 2

z = 2*(cos(90°) + isin(90°)), 2*(cos(210°) + isin(210°)), 2*(cos(330°) + isin(330°))

So, the solutions are: z = 2i, -1 + i√3, -1 - i√3

b) To solve for z, we need to find the 5th root of 1+i.

First, we convert 1+i to polar form:
1 + i = √2 * (cos(45°) + isin(45°))

Now, z = (√2)^(1/5)*(cos((45° + n*360°)/5) + isin((45° + n*360°)/5)), where n = 0, 1, 2, 3, 4

z = ( ∜2 {cos(9°) + isin(9°), cos(81°) + isin(81°), cos(153°) + isin(153°), cos(225°) + isin(225°), cos(297°) + isin(297°)})

So, the solutions are: z = 1(cos(9°) + isin(9°)), 1(cos(81°) + isin(81°)), 1(cos(153°) + isin(153°)), 1(cos(225°) + isin(225°)), 1(cos(297°) + isin(297°))