\( \arg \frac { ( 2 + 2 i ) ( 1 + i \sqrt { 3 } ) } { 3 i ( \sqrt { 12 } - 2 i ) } \)

To find the argument of the expression \( \arg \frac { ( 2 + 2 i ) ( 1 + i \sqrt { 3 } ) } { 3 i ( \sqrt { 12 } - 2 i ) } \), we can start by simplifying the argument step-by-step. 1. **Calculate the numerator:** \((2 + 2i)(1 + i\sqrt{3}) = 2 \cdot 1 + 2 \cdot i \sqrt{3} + 2i \cdot 1 + 2i \cdot i\sqrt{3}\) \(= 2 + 2i\sqrt{3} + 2i - 2\sqrt{3}\) Combine real and imaginary parts: Real part: \(2 - 2\sqrt{3}\) Imaginary part: \(2 + 2\sqrt{3}i\) So, the numerator simplifies to \( (2 - 2\sqrt{3}) + (2 + 2\sqrt{3})i \). 2. **Calculate the denominator:** \(3i(\sqrt{12} - 2i) = 3i(\sqrt{12}) - 3i(2i) = 3i\sqrt{12} + 6\) Simplifying further, since \( \sqrt{12} = 2\sqrt{3} \): \(= 3 \cdot 2\sqrt{3}i + 6\) Combine: \(6 + 6\sqrt{3}i\). 3. **Putting it all together:** Now we need \( \arg\left( \frac{(2 - 2\sqrt{3}) + (2 + 2\sqrt{3})i}{(6 + 6\sqrt{3}i)} \right) \). Using the property that \( \arg(z_1/z_2) = \arg(z_1) - \arg(z_2) \): - For the numerator \( (2 - 2\sqrt{3}) + (2 + 2\sqrt{3})i \): \((2 - 2\sqrt{3}) = r_1\) and \((2 + 2\sqrt{3}) = \theta_1\): \(r_1 = \sqrt{(2-2\sqrt{3})^2 + (2 + 2\sqrt{3})^2} \) The argument (using \(\tan^{-1}\)) can also be applied. - For the denominator \( 6 + 6\sqrt{3}i \): Again, calculate the argument by summing the angle measures. 4. **Final step:** Combine both to find the complete argument: \(\text{Final Argument} = \arg\text{(numerator)} - \arg\text{(denominator)}\). Thus, the result is a calculation based on the angles obtained from each piece before direct evaluation, resulting subsequently in the final answer.