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The expression given is:

\[
\sqrt{-7 - 24i}
\]

We are asked to write this in the form \(a + bi\), where \(a\) and \(b\) are real numbers. This involves expressing the square root of a complex number in its standard form.

To solve this:

### Step 1: Express \( -7 - 24i \) in polar form.
First, we calculate the modulus \(r\) and the argument \(\theta\) of the complex number \( -7 - 24i \).

- The modulus \(r\) is given by:

\[
r = \sqrt{(-7)^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25
\]

- The argument \(\theta\) is the angle formed with the real axis, calculated as:

\[
\theta = \tan^{-1}\left(\frac{-24}{-7}\right)
\]
This simplifies to:

\[
\theta = \tan^{-1}\left(\frac{24}{7}\right)
\]

Using a calculator, \(\theta \approx 1.2925\) radians. Since the complex number is in the third quadrant, the actual argument is:

\[
\theta = \pi + 1.2925 \approx 4.4341 \, \text{radians}
\]

### Step 2: Apply the square root in polar form.
The square root of a complex number in polar form \(re^{i\theta}\) is:

\[
\sqrt{r}e^{i\frac{\theta}{2}}
\]

- \(\sqrt{r} = \sqrt{25} = 5\)
- The new argument is:

\[
\frac{\theta}{2} = \frac{4.4341}{2} \approx 2.217
\]

Thus, the square root in polar form is:

\[
5 \left( \cos(2.217) + i\sin(2.217) \right)
\]

### Step 3: Convert back to rectangular form.
Now we calculate the cosine and sine of \(2.217\):

\[
\cos(2.217) \approx -0.607 \quad \text{and} \quad \sin(2.217) \approx 0.794
\]

Therefore, the square root in rectangular form is:

\[
5(-0.607 + 0.794i) = -3.035 + 3.97i
\]

### Final Answer:
The expression \( \sqrt{-7 - 24i} \) in the form \(a + bi\) is approximately:

\[
\boxed{-3.04 + 3.97i}
\]

Would you like any additional details on this process?

Here are five related questions:
1. How do you express a complex number in polar form?
2. What is the general formula for taking the square root of a complex number?
3. How does the argument of a complex number change based on its quadrant?
4. What is the relationship between polar and rectangular forms of complex numbers?
5. How can we apply De Moivre's theorem in operations involving complex numbers?

**Tip:** When calculating the argument \(\theta\) for a complex number, always check the quadrant to adjust the angle correctly.

Write the expression \(\sqrt{-7 - 24i}\) in the form \(a + bi\) where \(a\) and \(b\) are real numbers.

Learn how to express the complex number \(\sqrt{-7 - 24i}\) in the form \(a + bi\) using polar form and De Moivre's Theorem. Includes detailed steps for calculating modulus, argument, and converting back to rectangular form. Suitable for Grades 11-12.

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