Question

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

(i) $\frac{11}{\left(2^3\times 3\right)}$
(ii) $\frac{73}{\left(2^2\times 3^3\times 5\right)}$
(iii)$\frac{129}{\left(2^2\times 5^7\times 7^5\right)}$
(iv)$\frac{9}{35}$
(v)$\frac{77}{210}$
(vi) $\frac{32}{147}$
(vii) $\frac{29}{343}$
(viii) $\frac{64}{455}$

Answer 1

(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, $(2^3\times 3)$≠ $(2^m\times 5^n)$
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, $(2^2\times 3^3\times 5)$ ≠ $(2^m\times 5^n)$
Hence, the given rational is non-terminating repeating decimal.

(iii) $\frac{129}{2^2\times 5^7\times 7^5}$
We know 2, 5 or 7 is not a factor of 129, so it is in its simplest form.
Moreover, $(2^2\times 5^7\times 7^5)$ ≠ $(2^m\times 5^n)$
Hence, the given rational is non-terminating repeating decimal.

(iv) $\frac{9}{35}=\frac{9}{5\times 7}$
We know either 5 or 7 is not a factor of 9, so it is in its simplest form.
Moreover, ($5\times 7$) ≠ $(2^m\times 5^n)$
Hence, the given rational is non-terminating repeating decimal.


(v) $\frac{77}{210}=\frac{77÷7}{210÷7}=\frac{11}{30}=\frac{11}{2\times 3\times 5}$
We know 2, 3 or 5 is not a factor of 11, so $\frac{11}{30}$ is in its simplest form.
Moreover, $(2\times 3\times 5)$ ≠ $(2^m\times 5^n)$
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, $(3\times 7^2)$ ≠ $(2^m\times 5^n)$
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, $7^3$ ≠ $(2^m\times 5^n)$
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, $(5\times 7\times 13)$ ≠ $(2^m\times 5^n)$
Hence, the given rational is non-terminating repeating decimal.