Given the modulus $|z| = 2$ and the argument $\arg(z) = -\frac{2}{3}\pi$, we can express $z$ in its polar form using the following formula:

$z = |z| \left( \cos(\arg(z)) + i \sin(\arg(z)) \right)$

Substituting the given values:

$z = 2 \left( \cos\left(-\frac{2}{3}\pi\right) + i \sin\left(-\frac{2}{3}\pi\right) \right)$

Step 1: Calculate $\cos\left(-\frac{2}{3}\pi\right)$

Using the cosine function: $\cos\left(-\frac{2}{3}\pi\right) = \cos\left(\frac{2}{3}\pi\right) = -\frac{1}{2}$

Step 2: Calculate $\sin\left(-\frac{2}{3}\pi\right)$

Using the sine function: $\sin\left(-\frac{2}{3}\pi\right) = -\sin\left(\frac{2}{3}\pi\right) = -\frac{\sqrt{3}}{2}$

Step 3: Write the complex number in polar form

Now, substituting these values into the formula: $z = 2 \left( -\frac{1}{2} + i \left( -\frac{\sqrt{3}}{2} \right) \right)$ Simplifying: $z = -1 - i\sqrt{3}$

Thus, the complex number in standard form is: $z = -1 - i\sqrt{3}$

Would you like any further clarifications or details?

Here are five related questions for further exploration:

1. How do you convert from polar form to exponential form of a complex number?
2. What is the geometric interpretation of the argument of a complex number?
3. How do you multiply two complex numbers in polar form?
4. How do you find the argument of a complex number located in the third or fourth quadrant?
5. What is the relation between Cartesian form and polar form of complex numbers?

Tip: When working with complex numbers, always be mindful of the quadrant to correctly determine the sign of sine and cosine.