Determine whether the rational number frac{77 | vedclass

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(B) First,simplify the fraction $\frac{77}{210}$ by dividing both numerator and denominator by their greatest common divisor,which is $7$.$\frac{77 \d

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Non-terminating repeating,$0.3666...$

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(B) First,simplify the fraction $\frac{77}{210}$ by dividing both numerator and denominator by their greatest common divisor,which is $7$.$\frac{77 \div 7}{210 \div 7} = \frac{11}{30}$.Next,find the prime factorization of the denominator $30$:$30 = 2 	imes 3 	imes 5$.$A$ rational number has a terminating decimal expansion if and only if the prime factorization of its denominator (in simplest form) is of the form $2^n 	imes 5^m$,where $n$ and $m$ are non-negative integers.Since the denominator $30$ contains a factor of $3$ (which is not $2$ or $5$),the decimal expansion of $\frac{11}{30}$ is non-terminating and repeating.Performing the division $11 \div 30$ gives $0.3666...$ or $0.3\overline{6}$.

Determine whether the rational number $\frac{77}{210}$ has a terminating or non-terminating repeating decimal expansion. If it is terminating,write down its decimal expansion.

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