Use the properties of the operations of rational number arithmetic to perform these calculations. Express your answers in simplest form. \( \begin{array}{ll}\text { (a) } \frac{2}{3} \cdot \frac{4}{7}+\frac{2}{3} \cdot \frac{3}{7} & \text { (b) } \frac{2}{3} \cdot \frac{2}{5}-\frac{1}{6} \cdot \frac{2}{5} \\ \text { (a) } \frac{2}{3} \cdot \frac{4}{7}+\frac{2}{3} \cdot \frac{3}{7}=\square \text { (Type an integer or a simplified fraction.) }\end{array} \)

Let's break down the calculations step by step! For part (a): \[ \frac{2}{3} \cdot \frac{4}{7} + \frac{2}{3} \cdot \frac{3}{7} = \frac{2}{3} \left( \frac{4}{7} + \frac{3}{7} \right) \] Now, combine the fractions inside the parentheses: \[ \frac{4}{7} + \frac{3}{7} = \frac{4 + 3}{7} = \frac{7}{7} = 1 \] So we have: \[ \frac{2}{3} \cdot 1 = \frac{2}{3} \] Thus, the answer for (a) is: \[ \frac{2}{3} \] For part (b): \[ \frac{2}{3} \cdot \frac{2}{5} - \frac{1}{6} \cdot \frac{2}{5} = \frac{2}{5} \left( \frac{2}{3} - \frac{1}{6} \right) \] Now, we need a common denominator to combine \(\frac{2}{3}\) and \(\frac{1}{6}\). The least common denominator is 6: \[ \frac{2}{3} = \frac{4}{6} \] So, \[ \frac{4}{6} - \frac{1}{6} = \frac{4 - 1}{6} = \frac{3}{6} = \frac{1}{2} \] Now substitute this back in: \[ \frac{2}{5} \cdot \frac{1}{2} = \frac{2 \cdot 1}{5 \cdot 2} = \frac{2}{10} = \frac{1}{5} \] Thus, the answer for (b) is: \[ \frac{1}{5} \] To summarize: (a) \(=\frac{2}{3}\) (b) \(=\frac{1}{5}\)