To determine which fractions can be represented as terminating decimals, we need to check the prime factors of their simplified denominators. A fraction can be represented as a terminating decimal if and only if, after simplification, the only prime factors of the denominator are 2 and/or 5.
Let's analyze each fraction one by one:
a. [tex]\(\frac{1}{5}\)[/tex]:
- The denominator is 5, which has a single prime factor of 5.
- Since the denominator is a power of 5, [tex]\(\frac{1}{5}\)[/tex] can be represented as a terminating decimal.
b. [tex]\(\frac{17}{40}\)[/tex]:
- The denominator is 40. We factorize it: [tex]\(40 = 2^3 \times 5\)[/tex].
- Since the denominator has only the prime factors 2 and 5, [tex]\(\frac{17}{40}\)[/tex] can be represented as a terminating decimal.
c. [tex]\(\frac{7}{14}\)[/tex]:
- We simplify this fraction. The greatest common divisor (GCD) of 7 and 14 is 7. So, [tex]\(\frac{7}{14}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex].
- The simplified denominator is 2, which is a power of 2.
- Therefore, [tex]\(\frac{7}{14}\)[/tex] can be represented as a terminating decimal.
d. [tex]\(\frac{1}{2^7}\)[/tex]:
- The denominator is [tex]\(2^7 = 128\)[/tex], which is a power of 2.
- Since the denominator has only the prime factor 2, [tex]\(\frac{1}{128}\)[/tex] can be represented as a terminating decimal.
e. [tex]\(\frac{9}{51}\)[/tex]:
- We simplify this fraction. The greatest common divisor (GCD) of 9 and 51 is 3. So, [tex]\(\frac{9}{51}\)[/tex] simplifies to [tex]\(\frac{3}{17}\)[/tex].
- The simplified denominator is 17, which is a prime number not equal to 2 or 5.
- Therefore, [tex]\(\frac{9}{51}\)[/tex] cannot be represented as a terminating decimal.
f. [tex]\(\frac{119}{625}\)[/tex]:
- The denominator is 625. We factorize it: [tex]\(625 = 5^4\)[/tex].
- Since the denominator has only the prime factor 5, [tex]\(\frac{119}{625}\)[/tex] can be represented as a terminating decimal.
Based on the above analysis, the fractions that can be represented as terminating decimals are:
- [tex]\(\frac{1}{5}\)[/tex]
- [tex]\(\frac{17}{40}\)[/tex]
- [tex]\(\frac{7}{14}\)[/tex]
- [tex]\(\frac{1}{2^7}\)[/tex]
- [tex]\(\frac{119}{625}\)[/tex]
Thus, the fractions [tex]\(\frac{1}{5}\)[/tex], [tex]\(\frac{17}{40}\)[/tex], [tex]\(\frac{7}{14}\)[/tex], [tex]\(\frac{1}{2^7}\)[/tex], and [tex]\(\frac{119}{625}\)[/tex] can all be represented as terminating decimals, whereas [tex]\(\frac{9}{51}\)[/tex] cannot.
