To compute the square root of the complex number $-4 - 2i$, we follow these steps:

Step 1: Convert to polar form

First, express the complex number in polar form. A complex number $z = a + bi$ can be written in polar form as:

$z = r \left( \cos \theta + i \sin \theta \right)$

Where:

• $r = \sqrt{a^2 + b^2}$ (the modulus),
• $\theta = \text{atan2}(b, a)$ (the argument or phase angle).

For $-4 - 2i$, we have $a = -4$ and $b = -2$.

Step 2: Calculate the modulus

$r = \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$

Step 3: Calculate the argument

$\theta = \text{atan2}(-2, -4)$

Since both the real and imaginary parts are negative, the argument will be in the third quadrant. We can calculate:

$\theta = \tan^{-1}\left(\frac{-2}{-4}\right) = \tan^{-1}(0.5)$

The exact angle $\tan^{-1}(0.5)$ is approximately $26.565^\circ$, but since the complex number is in the third quadrant, we add $180^\circ$, so:

$\theta \approx 180^\circ + 26.565^\circ = 206.565^\circ$

Or in radians:

$\theta \approx 3.605 \, \text{radians}$

Step 4: Apply the square root formula

The square root of a complex number in polar form is given by:

$\sqrt{r} \left( \cos \frac{\theta}{2} + i \sin \frac{\theta}{2} \right)$

Substituting $r = 2\sqrt{5}$ and $\theta \approx 3.605 \, \text{radians}$:

1. Compute $\sqrt{r} = \sqrt{2\sqrt{5}} = \sqrt{2} \cdot \sqrt[4]{5}$.
2. Divide $\theta$ by 2: $\frac{3.605}{2} \approx 1.8025$.

Thus, the square root of $-4 - 2i$ is approximately:

$\sqrt{2\sqrt{5}} \left( \cos 1.8025 + i \sin 1.8025 \right)$

This can now be simplified further for an exact or numerical solution. Would you like to continue simplifying or use numerical values for a final answer?

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Follow-up Questions:

1. Would you like to see the detailed numerical evaluation of the square root?
2. Do you want an explanation of polar coordinates and their use in complex numbers?
3. Should we cover how to convert back to rectangular form after using polar form?
4. Would you like help visualizing this operation geometrically?
5. Do you have questions about how angles and quadrants work with complex numbers?

Tip: Using polar form makes operations like multiplication, division, and finding roots much easier than in rectangular form!