1+i+1/3+корень3/3i

To simplify the expression \( 1 + i + \frac{1}{3} + \frac{\sqrt{3}}{3} i \), we can first gather the real and imaginary parts separately.

Step 1: Gather Real Parts

• The real parts are \( 1 + \frac{1}{3} \).

Calculating the Real Part: \[ 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]

Step 2: Gather Imaginary Parts

• The imaginary parts are \( i + \frac{\sqrt{3}}{3} i \).

Calculating the Imaginary Part: \[ i + \frac{\sqrt{3}}{3} i = \left(1 + \frac{\sqrt{3}}{3}\right) i \]

Step 3: Write the Combined Result Now we can combine the results: \[ 1 + i + \frac{1}{3} + \frac{\sqrt{3}}{3} i = \frac{4}{3} + \left(1 + \frac{\sqrt{3}}{3}\right)i \]

Thus, the simplified expression is: \[ \frac{4}{3} + \left(1 + \frac{\sqrt{3}}{3}\right)i \]