Without actually performing the long division, find if \(\dfrac{987}{10500}\) will have terminating or non-terminating decimal expansion. Give reasons.
Terminating decimal expansion, because denominator reduces to \(2^2 \times 5^3\).
Step 1: Start with the given fraction: \(\dfrac{987}{10500}\).
Step 2: Divide numerator and denominator by their common factor 3: \(\dfrac{987}{10500} = \dfrac{329}{3500}\).
Step 3: Now, divide numerator and denominator again by 7:
\(\dfrac{329}{3500} = \dfrac{47}{500}\).
Step 4: Factorize the denominator 500. We get \(500 = 2^2 \times 5^3\).
Step 5: Since the denominator in lowest terms has only the prime factors 2 and 5, the decimal expansion must be terminating.
Final Conclusion: The fraction \(\dfrac{987}{10500}\) has a terminating decimal expansion.