Calculate the solutions to
$$\left(-8-8\sqrt{3}i\right)^3$$
I would really appreciate if you could help me with this. Thanks
$\endgroup$ 23 Answers
$\begingroup$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 $$(a + b)^3 = (a + b)(a+b)(a+b) = a^3 + 3a^2 b + 3ab^2 + b^3$$
$\endgroup$ $\begingroup$An idea using polar coordinates:
$$-8-8\sqrt3\,i=16\left(-\frac12-\frac{\sqrt3}2i\right)=16e^{\frac{4\pi i}3}\implies$$
$$(-8-8\sqrt3\,i)^3=16^3e^{4\pi i}=\ldots$$
$\endgroup$ $\begingroup$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.
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