If` z_1=sqrt2 (cos 30^(@) +isin60^(@)),z_2=sqrt3 (cos 60^(@) +isin30^(@)) `Find` R_e(z_1z_2)`

Answer

To solve the problem, we need to find the real part of the product of two complex numbers \( z_1 \) and \( z_2 \). Let's break down the solution step by step.

### Step 1: Write down the complex numbers
Given:
\[ z_1 = \sqrt{2} \left( \cos 30^\circ + i \sin 30^\circ \right) \]
\[ z_2 = \sqrt{3} \left( \cos 60^\circ + i \sin 30^\circ \right) \]

### Step 2: Substitute the values of cosine and sine
Using the known values:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}, \quad \cos 60^\circ = \frac{1}{2}, \quad \sin 60^\circ = \frac{\sqrt{3}}{2} \]

Substituting these values into \( z_1 \) and \( z_2 \):
\[ z_1 = \sqrt{2} \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = \frac{\sqrt{6}}{2} + i \frac{\sqrt{2}}{2} \]
\[ z_2 = \sqrt{3} \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) = \frac{\sqrt{3}}{2} + i \frac{3}{2} \]

### Step 3: Multiply \( z_1 \) and \( z_2 \)
Now, we will multiply \( z_1 \) and \( z_2 \):
\[ z_1 z_2 = \left( \frac{\sqrt{6}}{2} + i \frac{\sqrt{2}}{2} \right) \left( \frac{\sqrt{3}}{2} + i \frac{3}{2} \right) \]

Using the distributive property (FOIL):
\[ = \frac{\sqrt{6}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{6}}{2} \cdot i \frac{3}{2} + i \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + i \frac{\sqrt{2}}{2} \cdot i \frac{3}{2} \]

### Step 4: Simplify the multiplication
Calculating each term:
1. \( \frac{\sqrt{6} \cdot \sqrt{3}}{4} = \frac{\sqrt{18}}{4} = \frac{3\sqrt{2}}{4} \)
2. \( \frac{3\sqrt{6}}{4} i \)
3. \( \frac{\sqrt{6}}{4} i \)
4. \( -\frac{3\sqrt{2}}{4} \) (since \( i^2 = -1 \))

Combining these:
\[ z_1 z_2 = \left( \frac{3\sqrt{2}}{4} - \frac{3\sqrt{2}}{4} \right) + i \left( \frac{3\sqrt{6}}{4} + \frac{\sqrt{6}}{4} \right) \]
\[ = 0 + i \left( \frac{4\sqrt{6}}{4} \right) = 0 + i \sqrt{6} \]

### Step 5: Find the real part
The real part of \( z_1 z_2 \) is:
\[ R_e(z_1 z_2) = 0 \]

### Final Answer
Thus, the real part \( R_e(z_1 z_2) \) is:
\[ \boxed{0} \]