The decimal expansion of the rational number | vedclass

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Question

(D) To determine the number of decimal places after which the rational number $\frac{14587}{1250}$ terminates,we express the denominator in its prime

Vedclass · Accepted Answer

four decimal places

Vedclass · Answer

(D) To determine the number of decimal places after which the rational number $\frac{14587}{1250}$ terminates,we express the denominator in its prime factorization form.$1250 = 2^1 	imes 5^4$.The decimal expansion of a rational number $\frac{p}{q}$ terminates after $n$ decimal places,where $n$ is the maximum of the exponents of $2$ and $5$ in the prime factorization of the denominator $q$ (when the fraction is in its simplest form).Here,the exponents are $1$ and $4$. The maximum exponent is $4$.Alternatively,we can write:$\frac{14587}{1250} = \frac{14587}{2^1 	imes 5^4} = \frac{14587 	imes 2^3}{2^1 	imes 5^4 	imes 2^3} = \frac{14587 	imes 8}{2^4 	imes 5^4} = \frac{116696}{10^4} = \frac{116696}{10000} = 11.6696$.Since the result is $11.6696$,the decimal expansion terminates after four decimal places.

The decimal expansion of the rational number $\frac{14587}{1250}$ will terminate after

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