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The image shows a problem involving three complex numbers \( z_1 \), \( z_2 \), and \( z_3 \), which are represented on the complex plane. Based on their geometric representations, we can determine their exponential form (polar form). Here are the steps to express each number and solve the operations.

### Identifying the complex numbers:
1. **\( z_1 \)** is on the positive real axis at 2, so:
   \[
   z_1 = 2e^{i0^\circ} = 2
   \]

2. **\( z_2 \)** is in the first quadrant, with a modulus of \( 2\sqrt{3} \) and an argument of \( 60^\circ \), so:
   \[
   z_2 = 2\sqrt{3} e^{i60^\circ}
   \]

3. **\( z_3 \)** is in the third quadrant, with a modulus of \( 4\sqrt{3} \) and an argument of \( -120^\circ \), so:
   \[
   z_3 = 4\sqrt{3} e^{-i120^\circ}
   \]

Now, let’s perform the operations:

### a) \( \frac{z_1}{z_2} \)
Using the formula for division of complex numbers in exponential form:
\[
\frac{z_1}{z_2} = \frac{2}{2\sqrt{3}} e^{i(0^\circ - 60^\circ)} = \frac{1}{\sqrt{3}} e^{-i60^\circ}
\]

### b) \( \sqrt{z_1} \)
The square root of \( z_1 = 2 \) is:
\[
\sqrt{z_1} = \sqrt{2} e^{i0^\circ}
\]

### c) \( (z_2)^3 \)
To cube \( z_2 \), cube the modulus and multiply the argument by 3:
\[
(z_2)^3 = (2\sqrt{3})^3 e^{i(60^\circ \times 3)} = 24\sqrt{3} e^{i180^\circ}
\]
This is equal to \( -24\sqrt{3} \).

### d) \( z_3 \cdot z_2 \)
Multiply the moduli and add the arguments:
\[
z_3 \cdot z_2 = (4\sqrt{3})(2\sqrt{3}) e^{i(-120^\circ + 60^\circ)} = 24 e^{-i60^\circ}
\]

### e) \( \frac{z_2}{z_3} \)
Divide the moduli and subtract the arguments:
\[
\frac{z_2}{z_3} = \frac{2\sqrt{3}}{4\sqrt{3}} e^{i(60^\circ - (-120^\circ))} = \frac{1}{2} e^{i180^\circ} = -\frac{1}{2}
\]

Let me know if you want a deeper explanation of any step or have additional questions!

### Related questions:
1. How do you convert a complex number from Cartesian to polar form?
2. What is the geometric interpretation of multiplying two complex numbers in polar form?
3. How do the arguments of complex numbers change when performing division?
4. Can you explain the meaning of the argument of a complex number?
5. What is the significance of the modulus of a complex number in various operations?

#### Tip:
When multiplying or dividing complex numbers in polar form, the moduli multiply or divide, and the arguments add or subtract.

Perform operations with complex numbers z1, z2, and z3 represented on the complex plane and express the results in exponential form.

Learn how to perform operations on complex numbers z1, z2, and z3 represented on the complex plane. This problem involves key concepts in complex numbers, such as polar form and exponential notation, including operations like division, multiplication, and powers of complex numbers.

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