Question

​What can you say about the prime factorisations of the denominators of the following rationals:
(i) 43.123456789
(ii) $43.\overline{123456789}$
(iii) $27.\overline{142857}$
(iv) 0.120120012000120000 ...

Answer 1

Step-by-step explanation:

SOLUTION:

(i) Here, 43.123456789 has terminating decimal expansion. So,it represent a rational number .

43.123456789 =  43123456789/1000000000

= 43123456789/10^9

= 43123456789/ (2^9 × 5^9)

Its denominator is of the form 2m x 5n, where m, n are non-negative integers.

Denominator = 10^9 = (2^9 × 5^9)

 

(ii) Since ,43.123456789 (bar on 123456789) has non-terminating but repeating  decimal expansion. So it is a rational number.Its denominator has factors other than 2 or 5.

Let x = 43.123456789123456789….   ……… (1)

On multiplying by 1000000000 on both sides , we get  

1000000000 x = 43123456789.123456789 (bar on 123456789) ……….(2)

On Subtracting eq 1 from eq 2,  

999999999 x = 43123456746

x = 43123456746/999999999  

Hence, denominator = 999999999 which have factors other than 2 or 5.

 

(iii) Since 27.142857 (bar on 142857)  has non-terminating but repeating decimal expansion. So, its denominator has factors other than 2 or 5.

Hence, denominator = 999999 which have factors other than 2 or 5.

(iv) Since 0.120120012000120000 … has non-terminating non repeating decimal expansion. So, it is an irrational number.

HOPE THIS ANSWER WILL HELP YOU…

Answer 2

<b></b>HeYaA mAtE

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