# Hints on solving this equation

1. Apr 9, 2009

### missmerisha

1. The problem statement, all variables and given/known data

The solutions of the equation a^3 z^3+b^3 i =0
where a,b are an element of R+

3. The attempt at a solution
a^3 z^3 = - b^3 i
z^3 = (-b^3 i) / (a^3)

Now I'm stuck.
Any other suggestions?

2. Apr 9, 2009

Note that

$$i^3 = -i$$

As a start you can rewrite the entire right side as a 3rd power.

3. Apr 10, 2009

### Tedjn

You may also want to write z in terms of its real and imaginary components.

4. Apr 10, 2009

### missmerisha

That has nothing to do with the question.

5. Apr 10, 2009

### millitiz

Actually, it has. In fact, I think it is a genius way to solve the problem! Just making sure, we are asked to solve z in terms of a, b?
Really, try his way, and see what you can do next.
Because after doing that substitution, basically there is only one more step to find get the solution

6. Apr 11, 2009

### HallsofIvy

Staff Emeritus
It has everything to do with the equation. Assuming that a and b are real numbers, the cube root of $-a^3/b^3$ is -a/b so the only question is the cube root of i. Knowing that $i^3= -i$ helps with that. Warning: $-ia^3/b^3$
has three cube roots.

7. Apr 11, 2009

### arildno

Re-write the left-hand side as:
$$(az)^{3}-(ib)^{3}=0\to{(az-ib)((az)^{2}+iazb-b^{2})=0$$