​Which of the following rational numbers have a non-terminating decimal expansion? ​

Correct option is C

Concept Used:

A rational number has a non-terminating decimal expansion if its denominator, when prime factorized, contains prime factors other than 2 and 5.

Solution:

Option A: $$\frac{11}{1000}$$​.

The denominator 1000 has prime factors of 2 and 5 only (1000 = 2³ $$\times$$​5³).

Therefore, this fraction has a terminating decimal expansion.

​Option B: $$\frac{23}{2^35^2}$$ . 

The denominator 2³5² ( only contains 2 & 5)

Option C:$$\frac{19}{2^25^77^5}$$ .

The denominator 2²5⁷7⁵ ( contains 7, which is not 2 & 5)

Option D: $$\frac{4^2}{2^25^2}$$.

The denominator 2²5² ( contains 2 & 5)

Option (c) is right answer.