To find the value of $\sqrt{-3 - 4i}$, we'll express it in the polar form and then find the square root.

Step 1: Express the complex number in polar form

A complex number $z = a + bi$ can be expressed in polar form as: $z = r(\cos \theta + i\sin \theta)$ where: $r = \sqrt{a^2 + b^2}$ $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

For the given complex number $z = -3 - 4i$: $r = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ $\theta = \tan^{-1}\left(\frac{-4}{-3}\right) = \tan^{-1}\left(\frac{4}{3}\right)$

Since the complex number is in the third quadrant (both real and imaginary parts are negative), we need to adjust the angle: $\theta = \pi + \tan^{-1}\left(\frac{4}{3}\right)$

Step 2: Find the square root

The square root of a complex number $z$ in polar form $r(\cos \theta + i\sin \theta)$ is given by: $\sqrt{z} = \sqrt{r}\left(\cos\left(\frac{\theta}{2}\right) + i\sin\left(\frac{\theta}{2}\right)\right)$

Substituting the values: $\sqrt{r} = \sqrt{5}$ $\text{Angle } \frac{\theta}{2} = \frac{\pi + \tan^{-1}\left(\frac{4}{3}\right)}{2}$

Now, calculate: $\sqrt{-3 - 4i} = \sqrt{5}\left(\cos\left(\frac{\pi + \tan^{-1}\left(\frac{4}{3}\right)}{2}\right) + i\sin\left(\frac{\pi + \tan^{-1}\left(\frac{4}{3}\right)}{2}\right)\right)$

Step 3: Simplify the trigonometric expressions

Finally, we approximate the angles and simplify the expression to get the square root in the standard form $a + bi$.

The resulting complex number should be approximately: $\sqrt{-3 - 4i} \approx 1 - 2i$

Final Answer

$\sqrt{-3 - 4i} = 1 - 2i$

Would you like to dive deeper into any specific part of this process, or do you have any questions?

Here are five related questions for further exploration:

1. What is the geometric interpretation of finding the square root of a complex number?
2. How does the quadrant of the complex number affect the angle calculation?
3. How would you compute the square root of other complex numbers, like $3 + 4i$?
4. Can the square root of a complex number have more than one solution?
5. How is the polar form useful in complex number calculations?

Tip: When working with complex numbers, it's often helpful to first convert them into polar form, especially when performing operations like finding roots or multiplying.