To determine whether the product of the two fractions [tex]\(\frac{4}{109}\)[/tex] and [tex]\(\frac{16}{29}\)[/tex] represents a rational number or an irrational number, we can follow these steps:
1. Understanding Rational Numbers:
- A rational number is any number that can be expressed as the quotient or fraction of two integers, where the numerator (top number) and the denominator (bottom number) are both integers, and the denominator is not zero.
2. Product of Two Fractions:
- When we multiply two fractions, the product is found by multiplying the numerators together to get the new numerator and multiplying the denominators together to get the new denominator.
3. Apply the Multiplication:
- Multiply the numerators: [tex]\(4 \times 16 = 64\)[/tex]
- Multiply the denominators: [tex]\(109 \times 29 = 3161\)[/tex]
4. Form the Product:
- The product of the fractions [tex]\(\frac{4}{109}\)[/tex] and [tex]\(\frac{16}{29}\)[/tex] is [tex]\(\frac{64}{3161}\)[/tex].
5. Determine the Nature of the Result:
- The result [tex]\(\frac{64}{3161}\)[/tex] is expressed as a quotient of two integers where 64 is the integer numerator and 3161 is the integer denominator, and the denominator is non-zero.
Since the product [tex]\(\frac{64}{3161}\)[/tex] can be expressed as a ratio of two integers, it is by definition a rational number.
Conclusion:
The product [tex]\(\frac{4}{109} \times \frac{16}{29}\)[/tex] represents a rational number, as it can be expressed in the form [tex]\(\frac{64}{3161}\)[/tex], a ratio of two integers.
