cubing the expression of a complex number

Question

Calculate the solutions to

$$\left(-8-8\sqrt{3}i\right)^3$$

I would really appreciate if you could help me with this. Thanks

Answer 1

An idea using polar coordinates:

$$-8-8\sqrt3\,i=16\left(-\frac12-\frac{\sqrt3}2i\right)=16e^{\frac{4\pi i}3}\implies$$

$$(-8-8\sqrt3\,i)^3=16^3e^{4\pi i}=\ldots$$

Answer 2

Hint

$$\left(-8-8\sqrt{3}i\right)^3=-(16)^3 \left(\frac{1}{2}+i\frac{ \sqrt{3}}{2}\right)^3=-4096 \left(\cos \left(\frac{\pi }{3}\right)+i \sin \left(\frac{\pi }{3}\right)\right))^3$$

Use now de Moivre.

I am sure that you can take from here.

Answer 3

You can expand your expression, but cannot solve it, since there is no given equation with an unknown variable to solve for!

Now, to expand your expression, note that $$ \left(-8-8\sqrt{3}i\right)^3 = \Big((-8-8\sqrt 3 i)\times (-8-8\sqrt 3 i)\Big)\times(-8-8\sqrt 3i)$$

Note that we can compute the product $-8-8\sqrt 3 i)\times (-8-8\sqrt 3 i) $ just as we would compute any binomial $$(a + b)^2 = (a + b)(a+b) = a^2 + 2ab + b^2$$ and $$(a + b)^3 = (a + b)(a+b)(a+b) = a^3 + 3a^2 b + 3ab^2 + b^3$$