What type of decimal expansion will (\frac{14}{2\cdot 5^2\cdot 7^2}) have?

Non-terminating recurring

A rational number has a terminating decimal expansion after the fraction is reduced only when the denominator has no prime factors other than 2 and 5. If any other prime factor remains in the lowest terms, its decimal expansion is non-terminating but recurring. Therefore, cancellation must be performed before classifying the decimal; looking only at the original denominator could give a wrong conclusion.

Here, \(14=2\cdot7\), so cancellation with the numerator gives \(\frac{14}{2\cdot5^2\cdot7^2}=\frac{1}{5^2\cdot7}\). The reduced denominator still contains the prime factor 7. Hence the decimal cannot terminate and must be non-terminating recurring. Thus option B is correct. It is not non-recurring because every rational number has either a terminating or recurring decimal expansion.

This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Decimal expansion of rational numbers.