Answer :

(i) 7.2345

Here, 7.2345 has terminating decimal expansion.

So, it represents a rational number.

i.e. 7.2345 = =

Thus, q = 10^{4}, those factors are 2^{3} × 5^{3}

(ii)

is non-terminating but repeating.

So, it would be a rational number.

In a non-terminating repeating expansion of ,

q will have factors other than 2 or 5.

(iii) 23.245789

23.245789 is terminating decimal expansion

So, it would be a rational number.

i.e. 23.245789 = =

Thus, q = 10^{6}, those factors are 2^{5} × 5^{5}

In a terminating expansion of , q is of the form 2^{n}5^{m}

So, prime factors of q will be either 2 or 5 or both.

(iv)

is non-terminating but repeating.

So, it would be a rational number.

In a non-terminating repeating expansion of ,

q will have factors other than 2 or 5.

(v) 0.120120012000120000…

0.120120012000120000… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

(vi) 23.142857

23.142857 is terminating expansion.

So, it would be a rational number.

i.e. 23.142857 = =

Thus, q = 10^{6}, whose factors are 2^{5} × 5^{5}

In a terminating expansion of , q is of the form 2^{n}5^{m}

So, prime factors of q will be either 2 or 5 or both.

(vii) 2.313313313331…

2.313313313331… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

(viii) 0.02002000220002…

0.02002000220002… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

(ix) 3.300030000300003…

3.300030000300003… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

(x) 1.7320508…

1.7320508… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

(xi) 2.645713

2.645713 is terminating expansion

So, it would be a rational number.

i.e. 2.645713 = =

Thus, q = 10^{6}, those factors are 2^{5} × 5^{5}

In a terminating expansion of , q is of the form 2^{n}5^{m}

So, prime factors of q will be either 2 or 5 or both.

(xii) 2.8284271…

2.8284271… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

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