Evaluate. \[ \frac{7}{9}\left(\frac{1}{2}\right)^{2}+\frac{1}{4} \]

Let's evaluate the expression step by step: 1. Start with the first part: \[ \left(\frac{1}{2}\right)^{2} = \frac{1}{4} \] 2. Now substitute this back into the original expression: \[ \frac{7}{9}\left(\frac{1}{4}\right) + \frac{1}{4} \] 3. Next, calculate \(\frac{7}{9} \cdot \frac{1}{4}\): \[ \frac{7}{9} \cdot \frac{1}{4} = \frac{7}{36} \] 4. Now we have: \[ \frac{7}{36} + \frac{1}{4} \] 5. To add these fractions, we need a common denominator. The least common multiple of 36 and 4 is 36. Convert \(\frac{1}{4}\) to have a denominator of 36: \[ \frac{1}{4} = \frac{1 \cdot 9}{4 \cdot 9} = \frac{9}{36} \] 6. Now, we can add the fractions: \[ \frac{7}{36} + \frac{9}{36} = \frac{7 + 9}{36} = \frac{16}{36} \] 7. Finally, simplify \(\frac{16}{36}\): \[ \frac{16 \div 4}{36 \div 4} = \frac{4}{9} \] Thus, the final result is: \[ \boxed{\frac{4}{9}} \]