To simplify the given expression [tex]\(\left(1 + \frac{1}{3}\right)^2 - \frac{2}{9}\)[/tex], follow these steps carefully:
### Step 1: Convert the Mixed Number to an Improper Fraction
First, express the mixed number [tex]\(1 + \frac{1}{3}\)[/tex] as an improper fraction.
[tex]\[
1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}
\][/tex]
### Step 2: Square the Improper Fraction
Next, square the improper fraction [tex]\(\frac{4}{3}\)[/tex].
[tex]\[
\left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2} = \frac{16}{9}
\][/tex]
### Step 3: Subtract the Second Term from the Squared Value
Now, subtract [tex]\(\frac{2}{9}\)[/tex] from [tex]\(\frac{16}{9}\)[/tex].
[tex]\[
\frac{16}{9} - \frac{2}{9} = \frac{16 - 2}{9} = \frac{14}{9}
\][/tex]
### Conclusion
So, the expression [tex]\(\left(1 + \frac{1}{3}\right)^2 - \frac{2}{9}\)[/tex] simplifies to [tex]\(\frac{14}{9}\)[/tex].
Therefore, the simplified fraction is:
[tex]\[
\boxed{\frac{14}{9}}
\][/tex]
