To determine whether the product of two numbers is rational or irrational, it's important to look at the nature of each number involved in the multiplication.
### Definitions:
1. Rational Number: A number that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, with denominator \(q\) not equal to zero.
2. Irrational Number: A number that cannot be expressed as a simple fraction - it's decimal representation is non-repeating and non-terminating.
### Given:
1. Irrational Number: \(8.57396817 \ldots\)
This number is irrational because its decimal representation is non-terminating and non-repeating.
2. Rational Number: \(\frac{5}{8}\)
This number is rational because it can be expressed as the fraction of two integers, 5 and 8.
### Multiplication Rule:
- The product of a rational number and an irrational number is always irrational. This is because multiplying a non-repeating and non-terminating decimal (irrational number) by a fraction (rational number) will still yield a non-repeating and non-terminating decimal.
### Conclusion:
Since \(8.57396817 \ldots\) is an irrational number and \(\frac{5}{8}\) is a rational number, their product will be irrational.
Therefore, the product [tex]\(8.57396817 \ldots \times \frac{5}{8}\)[/tex] is [tex]\(\boxed{\text{irrational}}\)[/tex].
