Question

The decimal expansion of the rational number \[\frac{43}{2^4 \times 5^3}\] will terminate after how many places of decimals?

Answer 1

We have,

`43/(2^4xx5^3)`

Theorem states: 

Let `x= p/q` be a rational number, such that the prime factorization of q is of the form  `2^n xx 5^m`, where m and n are non-negative integers.

Then, x has a decimal expression which terminates after k places of decimals, where k is the larger of m and n.

This is clear that the prime factorization of the denominator is of the form `2^n xx 5^m`.

Hence, it has terminating decimal expansion which terminates after 4 places of decimals.