1+i+1/3+корень3/3i

To simplify the expression \( 1 + i + \frac{1}{3} + \frac{\sqrt{3}}{3}i \), we start by combining like terms, specifically the real and imaginary parts.

1. Combine the real parts: \[ 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]

2. Combine the imaginary parts: \[ i + \frac{\sqrt{3}}{3}i = \left(1 + \frac{\sqrt{3}}{3}\right)i = \left(\frac{3}{3} + \frac{\sqrt{3}}{3}\right)i = \frac{3 + \sqrt{3}}{3}i \]

Now we can combine the two parts:

\[ 1 + i + \frac{1}{3} + \frac{\sqrt{3}}{3}i = \frac{4}{3} + \frac{3 + \sqrt{3}}{3}i \]

So the final simplified expression is:

\[ \frac{4}{3} + \frac{3 + \sqrt{3}}{3}i \]