Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion

(i)

Factorize the denominator we get,
3125 = 5×5×5×5×5 = 5⁵
The denominator is of the form 5^m
Hence, the decimal expansion of  is terminating.
(ii)

Factorize the denominator we get,
8 = 2 × 2 ×2 = 2³
The denominator is of the form 2^m
Hence, the decimal expansion of  is terminating.

(iii)

Factorize the denominator we get,
455 = 5×7×13
Since, the denominator is not in the form of 2^m × 5ⁿ, and it also contains 7 and 13 as its factors,
Its decimal expansion will be non-terminating repeating.
(iv)

Factorize the denominator we get,
1600 = 2⁶×5²
The denominator is in the form 2^m × 5ⁿ
Hence, the decimal expansion of  is terminating.
(v)

Factorize the denominator we get,
343 = 7³
Since the denominator is not in the form of 2^m × 5ⁿ,  it has 7 as its factors.
So, the decimal expansion of  non-terminating repeating.

(vi)

The denominator is in the form 2^m×5ⁿ
Hence, the decimal expansion of  is terminating.
(vii)

Since, the denominator is not in the form of 2^m × 5ⁿ, as it has 7 in denominator.
So, the decimal expansion of   is non-terminating repeating.

(viii)
The denominator is in the form 5ⁿ
Hence, the decimal expansion of  is terminating.

(ix)

Factorize the denominator we get,
10 = 2×5
The denominator is in the form 2^m×5ⁿ
Hence, the decimal expansion of  is terminating.

(x)

Factorize the denominator we get,
30 = 2×3×5
Since the denominator is not in the form of 2^m × 5ⁿ, as it has 3 in denominator.
So, the decimal expansion of   is a non-terminating repeating.