To determine which of the given numbers has a terminating decimal expansion, we need to understand the properties of terminating decimals. A number has a terminating decimal expansion if, after simplifying the fraction to its lowest terms, its denominator has only the prime factors 2 and/or 5.
Let's analyze each option one by one:
1. Option A: [tex]\(\frac{2}{7}\)[/tex]
- The denominator is 7.
- The prime factorization of 7 is simply 7, which is neither 2 nor 5.
- Therefore, [tex]\(\frac{2}{7}\)[/tex] does not have a terminating decimal expansion.
2. Option B: [tex]\(\sqrt{12}\)[/tex]
- [tex]\(\sqrt{12}\)[/tex] is an irrational number because 12 is not a perfect square.
- Irrational numbers do not have a terminating or repeating decimal expansion.
- Therefore, [tex]\(\sqrt{12}\)[/tex] does not have a terminating decimal expansion.
3. Option C: [tex]\(\frac{5}{8}\)[/tex]
- The denominator is 8.
- The prime factorization of 8 is [tex]\(8 = 2^3\)[/tex], which consists only of the prime factor 2.
- Therefore, [tex]\(\frac{5}{8}\)[/tex] does have a terminating decimal expansion.
4. Option D: [tex]\(\frac{2}{11}\)[/tex]
- The denominator is 11.
- The prime factorization of 11 is simply 11, which is neither 2 nor 5.
- Therefore, [tex]\(\frac{2}{11}\)[/tex] does not have a terminating decimal expansion.
Based on this analysis, only [tex]\(\frac{5}{8}\)[/tex] has a terminating decimal expansion. Thus, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
