The problem asks to find all complex numbers $z$ such that:

$z^4 = 2 \left( \sqrt{3} + i \right)$

and express each solution in the form $z = a + bi$, where $a$ and $b$ are real numbers.

Step 1: Express the right-hand side in polar form

First, we express $2 (\sqrt{3} + i)$ in polar form. The general polar form of a complex number $x + yi$ is:

$r(\cos \theta + i \sin \theta)$

where:

• $r = |2(\sqrt{3} + i)|$ is the modulus (magnitude),
• $\theta = \arg(2(\sqrt{3} + i))$ is the argument (angle).

Find the modulus $r$:

$r = |2(\sqrt{3} + i)| = 2 \times \sqrt{ (\sqrt{3})^2 + 1^2 } = 2 \times \sqrt{3 + 1} = 2 \times \sqrt{4} = 2 \times 2 = 4$

Find the argument $\theta$:

The argument $\theta$ is the angle whose tangent is given by: $\tan \theta = \frac{1}{\sqrt{3}}$

This corresponds to an angle of $\frac{\pi}{6}$. So the polar form of $2 (\sqrt{3} + i)$ is:

$4 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)$

Step 2: Solve for $z^4$

We now have the equation in polar form:

$z^4 = 4 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)$

Taking the fourth roots of both sides, we get:

$z = 4^{1/4} \left( \cos \frac{\pi}{24} + i \sin \frac{\pi}{24} \right)$

We also account for the periodicity of the argument by adding multiples of $\frac{2\pi k}{4}$ for $k = 0, 1, 2, 3$.

Thus, the general solutions are:

$z_k = 4^{1/4} \left( \cos \frac{\pi + 2k\pi}{24} + i \sin \frac{\pi + 2k\pi}{24} \right) \quad \text{for } k = 0, 1, 2, 3$

Let me know if you'd like further details or a numerical approximation of the solutions!

Expansion questions:

1. How do you compute the modulus of any complex number?
2. What is the geometric meaning of taking the $n$-th root of a complex number?
3. Why is the argument of a complex number periodic?
4. Can you explain the polar form representation of complex numbers in detail?
5. What happens to the roots of unity when solving for powers of complex numbers?

Tip:

When working with complex numbers in polar form, always remember to account for the periodic nature of the argument (i.e., adding multiples of $2\pi$).