Determine the product of the given mixed numbers: $2 \frac{5}{8} \cdot 1 \frac{1}{7}$

Step 1 ：Convert the mixed numbers to improper fractions: \(2 \frac{5}{8} = \frac{21}{8}\) and \(1 \frac{1}{7} = \frac{8}{7}\)

Step 2 ：Multiply the two fractions: \(\frac{21}{8} \cdot \frac{8}{7} = \frac{168}{56}\)

Step 3 ：Simplify the fraction: \(\frac{168}{56} = \frac{21}{7} = 3\)

Step 4 ：The product of the given mixed numbers is \(\boxed{3}\)