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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.

Learn how to find the square root of the complex number -3 - 4i using polar form and trigonometry. This problem involves converting the complex number to polar form, calculating the square root in polar coordinates, and simplifying trigonometric expressions. Suitable for advanced high school students and those interested in complex number theory.

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