# Complex Numbers- Square root 3i

(1- sqrt 3i) ^3

I am having trouble solving the sqrt 3i part. I think I need to use de moivres theorem but I am unsure. If someone could push me in the right direction that would be a massive help. Thanks.

Related Calculus and Beyond Homework Help News on Phys.org
Cyosis
Homework Helper
What's the question exactly? You want to write that complex number in its polar representation or as x+iy? Either way start with writing $1-\sqrt{(3i)}$ in the polar representation. Once you have that it is easy to compute $(1-\sqrt{(3i)})^3$.

Last edited:
I think he meant:

$$\sqrt{1-i\sqrt{3}}$$

Use De Moivre's theorem to find all the solutions.

The question is:

If z= 5 +4i, write the number z(|z|^2-(1- sqrt3i)^3) in the form a+bi.

I am able to sub everything in but I want to simplify the (1- sqrt3i)^3. The square root has thrown me off.

Cyosis
Homework Helper
I take it the i is inside the square root? Try to write it in the form $3i=|z|e^{i\phi}$ then take the square root on both sides.

Is that meant to the theta or phi? Silly question I know. I have in my notes a simalar formula for Eulers formula but that uses theta not phi.... Another silly question I know but by changing the sq root 3i into a simple number going to give me the same result as getting the polar representation of the whole (1 - sqrt3i) ^3?

Cyosis
Homework Helper
It doesn't matter whether it's called phi, theta or JC_003 it is just a variable which represents the angle between |z| and the positive real axis. You will get the same answer both ways, as it should. However the angle theta is pretty hard to find for $1-\sqrt{3i}$. I suggest you write $\sqrt{3i}=x+iy$ first and then continue from there.

If this is an exercise from a text, I'd be very carefull about whether the question concerns

$$(\sqrt{3})i$$​

or

$$\sqrt{(3i)}$$​

Huge difference.

The question is not clear enough for me to help effectively, but I suspect -8 shows up somewhere.

--Elucidus