1(1)/(3)cdot (2)/(3)+(2)/(7):(3)/(2^2)+3=?

To solve the expression \(1\frac{1}{3} \cdot \frac{2}{3} + \frac{2}{7} : \frac{3}{2^2 + 3}\), we need to follow the order of operations (PEMDAS/BODMAS) and handle each part step by step.<br /><br />First, let's convert the mixed number \(1\frac{1}{3}\) to an improper fraction:<br />\[ 1\frac{1}{3} = \frac{4}{3} \]<br /><br />Now, we can rewrite the expression:<br />\[ \frac{4}{3} \cdot \frac{2}{3} + \frac{2}{7} : \frac{3}{2^2 + 3} \]<br /><br />Next, perform the multiplication:<br />\[ \frac{4}{3} \cdot \frac{2}{3} = \frac{4 \cdot 2}{3 \cdot 3} = \frac{8}{9} \]<br /><br />Now, let's simplify the denominator in the division part:<br />\[ 2^2 + 3 = 4 + 3 = 7 \]<br />So, the expression becomes:<br />\[ \frac{8}{9} + \frac{2}{7} : \frac{3}{7} \]<br /><br />Next, perform the division:<br />\[ \frac{2}{7} : \frac{3}{7} = \frac{2}{7} \cdot \frac{7}{3} = \frac{2 \cdot 7}{7 \cdot 3} = \frac{2}{3} \]<br /><br />Now, add the two fractions:<br />\[ \frac{8}{9} + \frac{2}{3} \]<br /><br />To add these fractions, we need a common denominator. The least common multiple of 9 and 3 is 9. Convert \(\frac{2}{3}\) to a fraction with a denominator of 9:<br />\[ \frac{2}{3} = \frac{2 \cdot 3}{3 \cdot 3} = \frac{6}{9} \]<br /><br />Now, add the fractions:<br />\[ \frac{8}{9} + \frac{6}{9} = \frac{8 + 6}{9} = \frac{14}{9} \]<br /><br />Therefore, the final answer is:<br />\[ \boxed{\frac{14}{9}} \]