To simplify the given expression [tex]\(\left(1+\frac{1}{3}\right)^2-\frac{2}{9}\)[/tex], we'll proceed step-by-step:
1. First, perform the addition inside the parentheses:
[tex]\[
1 + \frac{1}{3}
\][/tex]
Converting 1 to a fraction with a common denominator:
[tex]\[
1 = \frac{3}{3}
\][/tex]
Now add:
[tex]\[
\frac{3}{3} + \frac{1}{3} = \frac{4}{3}
\][/tex]
2. Next, square the result of this addition:
[tex]\[
\left(\frac{4}{3}\right)^2
\][/tex]
Calculate the square of [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[
\left(\frac{4}{3}\right)^2 = \frac{16}{9}
\][/tex]
3. Then, subtract [tex]\( \frac{2}{9} \)[/tex] from the squared result:
[tex]\[
\frac{16}{9} - \frac{2}{9}
\][/tex]
Since the denominators are the same, subtract the numerators:
[tex]\[
\frac{16}{9} - \frac{2}{9} = \frac{14}{9}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\left(1+\frac{1}{3}\right)^2-\frac{2}{9}\)[/tex] is:
[tex]\[
\boxed{\frac{14}{9}}
\][/tex]
