Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal.
(i)` 121/((2^(3)xx 3^(2)xx 7^(5)))` (ii) ` 17/90` (iii) ` 53/343` (iv) ` 66/180`

Explanation

Given number is ` 121/((2^(3)xx 3^(2)xx7^(5))) `
Clearly, none of 2,3 and 7 is a factor of 121.
so, the given rational number is in its simplest form.
And ` (2^(3) xx 3^(2) xx 7^(5)) ne ( 2^(m) xx 5^(n))`
` 121/ ((2^(3)xx 3^(2) xx 7^(5))` is a nonterminating repreating ldecimal.
(ii) Given numbers is ` 17/90`
And , ` 90= (2 xx 3^(2)xx 5)`
` Clearly, none of 2,3 and 4 is a factor of 17.
` 17/90` is in its simplest form.
` Also, ` 90= ( 2xx 3^(2)xx 5) ne ( 2^(m)xx 5^(n))`
` 17/90` is in its simplest form.
Also ` 90 = ( 2 xx 3^(2)xx 5) ne ( 2 ^(m) xx 5^(n))`
`17/90` is an nonterminating repeating decimal.
(iii) Given number is ` 53/343`
Now, ` 343= 7^(3) ` and 7 is not a factor of 53.
` 53/343` is in its simplest form.
Also, ` 343= 7^(3) ne ( 2^(m) xx 5^(n))`
` 53/343` is a nonterminating repeating decimal.
(iv) Given number is ` 66/180` and HCF ( 66 ,180) = 6
` 66/180 = ( 66 div 6)/(180 div 6) = 11/30`
Now , 30 = `( 2xx 3 xx 5) ` and none of 2,3,5 is a factor of 11.
` 11/30` is in its simplest form.
Also, ` 30 = ( 2xx 3 xx 5) ne (2^(m) xx 5^(n))`
`11/30 and "hence"66/180` is a nonterminating repeating decimal.