The complex conjugate of \( w=\sqrt{3}+i \) is denoted by \( w^{*} \). Solve \( 2 w+w^{*} \) Select one: a. \( 3 \sqrt{3}+3 i \) b. \( 3 \sqrt{3}+i \) c. \( \sqrt{3}+i \) d. \( \sqrt{3}+3 i \)

To solve \( 2w + w^{*} \), we first need to find the complex conjugate \( w^{*} \) of \( w = \sqrt{3} + i \). 1. **Find the complex conjugate**: The complex conjugate \( w^{*} \) is obtained by changing the sign of the imaginary part. Therefore: \[ w^{*} = \sqrt{3} - i \] 2. **Calculate \( 2w \)**: Now, we calculate \( 2w \): \[ 2w = 2(\sqrt{3} + i) = 2\sqrt{3} + 2i \] 3. **Add \( 2w \) and \( w^{*} \)**: Now we can add \( 2w \) and \( w^{*} \): \[ 2w + w^{*} = (2\sqrt{3} + 2i) + (\sqrt{3} - i) \] Combine the real and imaginary parts: \[ = (2\sqrt{3} + \sqrt{3}) + (2i - i) = 3\sqrt{3} + i \] Thus, the final result is: \[ 2w + w^{*} = 3\sqrt{3} + i \] The correct answer is: **b. \( 3\sqrt{3} + i \)**.