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Decimal Expansion of Rational and Irrational Numbers

Without actua...

Question

Without actual division , show that the following rational number is non-terminating repeating decimal : $\frac{64}{455}$

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Solution

Given rational number is $\frac{64}{455}$
Prime factorization of the denominator of the given rational number is
$455 = 5 \times 7 \times 13$
If the prime factorization of the denominator of any rational number is of the form $2^{m} \times 5^{n}$ where $m$ and $n$ are non-negative integers, then the decimal expansion of the given number is terminating else non-terminating repeating
$(5 \times 7 \times 13)$ is not of the form $(2^{m} \times 5^{n})$
Hence given fraction has non -terminating repeating decimal expansion.

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Similar questions

\frac{11}{(2^{3} \times 3)}

\frac{73}{(2^{2} \times 3^{3} \times 5)}

\frac{129}{(2^{2} \times 5^{7} \times 7^{5})}

\frac{9}{35}

\frac{77}{210}

\frac{32}{147}

\frac{29}{343}

\frac{64}{455}

\frac{11}{(2^{3} \times 3)}

\frac{73}{(2^{2} \times 3^{3} \times 5)}

\frac{129}{(2^{2} \times 5^{7} \times 7^{5})}

\frac{9}{35}

\frac{77}{210}

\frac{32}{147}

\frac{29}{343}

\frac{64}{455}

Without actual division, show that each of the following rational numbers is a non-terminating repeating decimal:

(iii) $\frac{9}{35}$

Without actual division, show that each of the following rational numbers is a non-terminating repeating decimal:

(v) $\frac{64}{455}$

Without actual division, show that each of the following rational numbers is a non-terminating repeating decimal:

(vii) $\frac{29}{343}$

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