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

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

We know either 2 or 3 is not a factor of 11, so it is in its simplest form.

Moreover, $\left(2^{3} \times 3\right) \neq\left(2^{m} \times 5^{n}\right)$

Hence, the given rational is non-terminating repeating decimal.

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

We know 2, 3 or 5 is not a factor of 73, so it is in its simplest form.

Moreover, $\left(2^{2} \times 3^{3} \times 5\right) \neq\left(2^{m} \times 5^{n}\right)$

Hence, the given rational is non-terminating repeating decimal.

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

We know either 5 or 7 is not a factor of 9, so it is in its simplest form.

Moreover, $(5 \times 7) \neq\left(2^{m} \times 5^{n}\right)$

Hence, the given rational is non-terminating repeating decimal.

(v) $\frac{77}{210}=\frac{77 \div 7}{210 \div 7}=\frac{11}{30}=\frac{11}{2 \times 3 \times 5}$

We know 2,3 or 5 is not a factor of 11 , so $\frac{11}{30}$ is in its simplest form.

Moreover, $(2 \times 3 \times 5) \neq\left(2^{m} \times 5^{n}\right)$

Hence, the given rational is non-terminating repeating decimal.

(vi) $\frac{32}{147}=\frac{32}{3 \times 7^{2}}$

 We know either 3 or 7 is not a factor of 32, so it is in its simplest form.

Moreover, $\left(3 \times 7^{2}\right) \neq\left(2^{m} \times 5^{n}\right)$

Hence, the given rational is non-terminating repeating decimal.

(vii) $\frac{29}{343}=\frac{29}{7^{3}}$

  We know 7 is not a factor of 29, so it is in its simplest form.

Moreover, $7^{3} \neq\left(2^{m} \times 5^{n}\right)$

Hence, the given rational is non-terminating repeating decimal.

(viii) $\frac{64}{455}=\frac{64}{5 \times 7 \times 13}$

We know 5, 7 or 13 is not a factor of 64, so it is in its simplest form.

Moreover, $(5 \times 7 \times 13) \neq\left(2^{m} \times 5^{n}\right)$

Hence, the given rational is non-terminating repeating decimal.