First, we find the probability that a single flower pot has a base diameter between 13.6 cm and 14.8 cm. We standardize the values using the normal distribution:

\(P(13.6 < X < 14.8) = P\left( \frac{13.6 - 14}{0.52} < z < \frac{14.8 - 14}{0.52} \right)\)

\(= P(-0.7692 < z < 1.538)\)

Using the standard normal distribution table, we find:

\(= \Phi(1.538) - [1 - \Phi(0.7692)]\)

\(= 0.9380 - [1 - 0.7791]\)

\(= 0.7171\)

Now, we use the binomial distribution to find the probability that exactly 8 out of 10 flower pots have diameters in this range:

\(P(8) = (0.7171)^8 (0.2829)^2 \binom{10}{8}\)

\(= 0.252\)