What type of decimal expansion will (\frac{200}{2^3\cdot 5^3\cdot 7}) have?

Non-terminating recurring

A rational number has a terminating decimal only when, after cancellation, its denominator has no prime factors other than 2 and 5. If another prime factor remains, the division cannot end; because remainders repeat, the decimal becomes non-terminating recurring. This rule helps classify the decimal without carrying out a long division.

Here, the numerator is 200 = \(2^3\times5^2\). Cancelling common factors with \(2^3\times5^3\times7\) leaves the denominator \(5\times7\), since one factor 5 and the factor 7 remain. The factor 7 is not allowed in a terminating denominator, so the decimal is non-terminating recurring. Therefore option B follows.

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