To calculate the square root of a complex number like \(-9 - 40i\), we can follow these steps:

### 1. Express the complex number in polar form:
First, write \(-9 - 40i\) in the form \( r(\cos \theta + i\sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument.

The modulus \( r \) is given by:
\[
r = \sqrt{(-9)^2 + (-40)^2} = \sqrt{81 + 1600} = \sqrt{1681} = 41
\]

The argument \( \theta \) is:
\[
\theta = \tan^{-1}\left(\frac{-40}{-9}\right) = \tan^{-1}\left(\frac{40}{9}\right)
\]

Calculating the angle:
\[
\theta \approx 77.32^\circ \text{ or } 1.3496 \text{ radians}
\]

Since \(-9 - 40i\) is in the third quadrant, we add \( \pi \) to the angle:
\[
\theta = \pi + 1.3496 \approx 4.4912 \text{ radians}
\]

### 2. Use De Moivre's theorem to find the square root:
The square roots of a complex number \( z = r(\cos \theta + i \sin \theta) \) are given by:
\[
\sqrt{z} = \sqrt{r}\left(\cos\left(\frac{\theta}{2}\right) + i \sin\left(\frac{\theta}{2}\right)\right)
\]

Substituting \( r = 41 \) and \( \theta = 4.4912 \):
\[
\sqrt{41} \approx 6.4031
\]
\[
\frac{\theta}{2} \approx \frac{4.4912}{2} \approx 2.2456 \text{ radians}
\]

Now calculate the real and imaginary parts:
\[
\cos(2.2456) \approx -0.623
\]
\[
\sin(2.2456) \approx 0.782
\]

Thus, the square root is approximately:
\[
\sqrt{-9 - 40i} \approx 6.4031 \left( -0.623 + 0.782i \right)
\]
\[
\sqrt{-9 - 40i} \approx -3.99 + 5.01i
\]

So, one of the square roots of \(-9 - 40i\) is approximately \(-3.99 + 5.01i\).

### Want more details or have questions?

Here are 5 related questions to expand your understanding:
1. How do you convert a complex number from rectangular form to polar form?
2. What is De Moivre's theorem, and how is it used with complex numbers?
3. How do you find the argument of a complex number in different quadrants?
4. What are the two square roots of a complex number, and why are there always two?
5. Can we express the square root of any negative real number in terms of complex numbers?

**Tip:** Always remember that a complex number's square root has two values, which differ by a sign in the imaginary or real part!

Learn how to calculate the square root of the complex number -9 - 40i using De Moivre's Theorem. This solution breaks down the process of converting the number into polar form, finding its modulus and argument, and applying the square root formula. Suitable for advanced high school and college students.

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