To solve the problem, we need to determine the modulus of the complex number $z = -4 - 4\sqrt{3}i$.

Modulus of a Complex Number

The modulus $|z|$ of a complex number $z = a + bi$ is given by: $|z| = \sqrt{a^2 + b^2}$ where $a$ is the real part and $b$ is the imaginary part.

In this case:

• $a = -4$ (real part),
• $b = -4\sqrt{3}$ (imaginary part).

Calculation of the Modulus

1. Square the real part $a$: $a^2 = (-4)^2 = 16$

2. Square the imaginary part $b$: $b^2 = (-4\sqrt{3})^2 = 16 \cdot 3 = 48$

3. Add the squares: $a^2 + b^2 = 16 + 48 = 64$

4. Take the square root to find the modulus: $|z| = \sqrt{64} = 8$

Answer

The modulus of the complex number $z = -4 - 4\sqrt{3}i$ is: $\boxed{8}$

Verification of the Options

From the given options:

• $16\sqrt{3}$ → Not correct.
• $\sqrt{28}$ → Simplifies to $2\sqrt{7}$ ≠ 8.
• $\sqrt{32}$ → Simplifies to $4\sqrt{2}$ ≠ 8.
• $8$ → Correct.

Do you want me to explain further or clarify any steps?

Related Questions

1. How do you find the modulus of a complex number geometrically?
2. What is the conjugate of a complex number $z = -4 - 4\sqrt{3}i$?
3. How do you plot complex numbers on the Argand plane?
4. How can you generalize the modulus formula for complex numbers?
5. What is the difference between a real and imaginary part of a complex number?

Tip:

Always square carefully when working with square roots in complex numbers to avoid small calculation errors.