To multiply the fractions \( \frac{3}{8} \) and \( \frac{4}{7} \), you multiply the numerators together and the denominators together:
\[ \frac{3}{8} \times \frac{4}{7} = \frac{3 \times 4}{8 \times 7} = \frac{12}{56} \]
Next, we simplify \( \frac{12}{56} \) by finding the greatest common divisor (GCD) of 12 and 56. The GCD is 4.
Now, divide both the numerator and the denominator by their GCD:
\[ \frac{12 \div 4}{56 \div 4} = \frac{3}{14} \]
So, the answer in simplest form is:
\[ \frac{3}{14} \]