https://file.aiquickdraw.com/mathbot/file/1723987963584czwb1nx1.jpg

To solve parts (a) and (b) of the problem, let's first carefully examine and address each part based on the image.

### Part (a): What is the value of \(\sqrt{-3 - 4i}\)?

We've already calculated this in a previous explanation:

\[
\sqrt{-3 - 4i} \approx 1 - 2i
\]

So the value of \(\sqrt{-3 - 4i}\) is \(1 - 2i\).

### Part (b): Verify that \( (1 - 2i)^2 = -3 - 4i \)

To verify that \( (1 - 2i)^2 = -3 - 4i \), we simply need to square the complex number \(1 - 2i\):

\[
(1 - 2i)^2 = (1 - 2i)(1 - 2i)
\]

Using the distributive property (FOIL method):

\[
(1 - 2i)(1 - 2i) = 1(1) + 1(-2i) + (-2i)(1) + (-2i)(-2i)
\]
\[
= 1 - 2i - 2i + 4i^2
\]

Since \(i^2 = -1\), we substitute:

\[
= 1 - 4i + 4(-1)
\]
\[
= 1 - 4i - 4
\]
\[
= -3 - 4i
\]

Thus, \( (1 - 2i)^2 = -3 - 4i \), which confirms that the calculation in part (a) was correct.

### Final Answer:
- **Part (a)**: The value of \(\sqrt{-3 - 4i}\) is \(1 - 2i\).
- **Part (b)**: Verification shows that \((1 - 2i)^2 = -3 - 4i\).

Would you like to explore this further or have any questions?

Here are five related questions to deepen your understanding:

1. What happens if you square the other root, \(-1 + 2i\), of \(-3 - 4i\)?
2. How can you find the square root of any general complex number?
3. What are the implications of complex roots in solving quadratic equations?
4. How does Euler's formula relate to the polar form of complex numbers?
5. Can you derive the square root of a complex number geometrically on the complex plane?

**Tip:** Always verify your solution by plugging it back into the original equation, especially with complex numbers, as it helps to ensure accuracy.

Learn how to solve the complex number problem involving \(\sqrt{-3 - 4i}\) and verify the equation \((1 - 2i)^2 = -3 - 4i\). This detailed solution covers the calculation of square roots of complex numbers and verification of complex equations, suitable for advanced high school students.

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