Write whether the rational number 7/75 will have a terminating decimal expansion or a nor-terminating repeating decimal expansion.

A terminating decimal is a decimal that ends. It must have finite number of digits after the decimal point.

While, a non-terminating decimal is a decimal that continues endlessly. For example: π

Also more theoretically, we can say is a terminating decimal if

(a) p and q are co-primes.

(b) q is of the form 2ⁿ5^m.

Here, p=7 and q=75.

Checking for co-primes:

Factors of 7 = 1, 7

Factors of 75 = 1, 3, 5, 15, 25, 75

Common factors of 7 and 75 = 1

⇒ 7 and 75 are co-primes.

[∵ two numbers are co-primes if the only number that divides them is 1]

And since, 75 = 3 × 5 × 5 ≠ 2ⁿ5^m

Thus, is non-terminating decimal.