To determine which of the given fractions can be expressed as a terminating decimal, we need to evaluate each fraction to see if its decimal representation ends after a finite number of digits.
A fraction in its simplest form can be expressed as a terminating decimal if and only if the denominator can be written as [tex]\(2^m \times 5^n\)[/tex], where [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are non-negative integers. This means the denominator should only contain the prime factors 2 and/or 5.
Let's examine each fraction:
1. [tex]\(\frac{2}{3}\)[/tex]
- The denominator is 3. The prime factors of 3 are not 2 or 5.
- Therefore, [tex]\(\frac{2}{3}\)[/tex] is not a terminating decimal.
2. [tex]\(\frac{3}{11}\)[/tex]
- The denominator is 11. The prime factors of 11 are not 2 or 5.
- Therefore, [tex]\(\frac{3}{11}\)[/tex] is not a terminating decimal.
3. [tex]\(\frac{5}{8}\)[/tex]
- The denominator is 8. The prime factors of 8 are [tex]\(2^3\)[/tex].
- Since 8 can be written as [tex]\(2^3\)[/tex] and does not require any factors of 5, it fits the criterion.
- Therefore, [tex]\(\frac{5}{8}\)[/tex] is a terminating decimal.
4. [tex]\(\frac{1}{9}\)[/tex]
- The denominator is 9. The prime factors of 9 are not 2 or 5, since 9 is [tex]\(3^2\)[/tex].
- Therefore, [tex]\(\frac{1}{9}\)[/tex] is not a terminating decimal.
By this examination, we conclude that the fraction [tex]\(\frac{5}{8}\)[/tex] can be expressed as a terminating decimal.
