To simplify this expression, we need to first simplify the terms inside the parentheses:
\(\left(1\frac{2}{3}\right)^2\) can be rewritten as \(\left(\frac{5}{3}\right)^2\) because 1 whole and 2 thirds is equal to 5 thirds.
Similarly, \(\left(\frac{2}{3}\right)^2\) can be simplified to \(\frac{4}{9}\).
Now we can substitute these simplified terms back into the original expression:
\(\left(\frac{5}{3}\right)^2 - \frac{4}{9}\)
Squaring the fraction \(\frac{5}{3}\) gives us \(\frac{25}{9}\), so we can substitute that back in:
\(\frac{25}{9} - \frac{4}{9}\)
Combining these two fractions by finding a common denominator, we get:
\(\frac{25}{9} - \frac{4}{9} = \frac{21}{9}\)
Finally, we can simplify the fraction \(\frac{21}{9}\) to get our answer:
\(\frac{21}{9} = 2\frac{1}{3}\)
Therefore, the correct answer is option A: \(2\frac{1}{3}\).