CBSE · Class 10 · Mathematics · Real NumbersWithout actually performing the long division, state whether the rational number 13/3125 has a terminating or a non-terminating repeating decimal expansion.

To determine whether the rational number 13/3125 has a terminating decimal expansion without actual division, we inspect the denominator 3125. First, we find the prime factorization of 3125, which is 5⁵. A rational number p/q has a terminating decimal expansion if the prime factorization of the denominator q is of the form 2ⁿ5ᵐ, where n and m are non-negative integers. Since 3125 can be written as 2⁰ × 5⁵, it satisfies this condition. Therefore, the given rational number has a terminating decimal expansion.