Hints on solving this equation

[missmerisha]

Homework Statement

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

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?

 

[statdad]

Note that

<br /> i^3 = -i<br />

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

 

[Tedjn]

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

 

[missmerisha]

Note that

<br /> i^3 = -i<br />

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

That has nothing to do with the question.

 

[millitiz]

That has nothing to do with the question.

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

 

[HallsofIvy]

That has nothing to do with the question.

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.

 

[arildno]

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