Question

Write the denominator of the rational number `257/5000` in the form 2^m × 5ⁿ, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.

Answer 1

Denominator of the rational number `257/5000` is 5000.

Now, 5000 = 2 × 2 × 2 × 5 × 5 × 5 × 5

= (2)³ × (5)⁴, which is of the type 2^m × 5ⁿ where m = 3 and n = 4 are non-negative integers.

∴ `257/5000 = 257/(2^3 xx 5^4) xx 2/2`  ...[Multiplying numerator and denominator by 2]

 = `514/(2^4 xx 5^4)`

= `514/(10)^4`

= `514/10000`

= 0.0514

Hence, 0.0514 is the required decimal expansion of the rational number `257/5000` and it is also a terminating decimal number.