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Question
The decimal expansion of the rational number \[\frac{43}{2^4 \times 5^3}\] will terminate after how many places of decimals?
Solution
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^nxx 5^m`, where mand n are non-negative integers.
Then, x has a decimal expression which terminates after k places of decimals, where k is the larger of mand n.
This is clear that the prime factorization of the denominator is of the form `2^nxx 5^m`,.
Hence, it has terminating decimal expansion which terminates after 4 places of decimal.
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