Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal.

$(i) \frac{11}{(2^{3} \times 3)} (i i) \frac{73}{(2^{2} \times 3^{3} \times 5)}$

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Solution

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

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

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

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

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

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Q. Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal.

(i) \frac{29}{343} (i i) \frac{64}{455}

Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal.

$(i) \frac{11}{(2^{3} \times 3)} (i i) \frac{73}{(2^{2} \times 3^{3} \times 5)} (i i i) \frac{129}{(2^{2} \times 5^{3} \times 7^{2})} (i v) \frac{9}{35} (v) \frac{77}{210} (v i) \frac{32}{147} (v i i) \frac{29}{343} (v i i i) \frac{64}{455}$

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

(i)

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

(ii)

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

(iii)

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

(iv)

\frac{9}{35}

(v)

\frac{77}{210}

(vi)

\frac{32}{147}

(vii)

\frac{29}{343}

(viii)

\frac{64}{455}

Q. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

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Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion : $\frac{23}{2^{3} {.5}^{2}}$

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Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal. (i) 11(23×3) (ii) 73(22×33×5)

Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal.

$(i) \frac{11}{(2^{3} \times 3)} (i i) \frac{73}{(2^{2} \times 3^{3} \times 5)}$