Question

Classift the following numbers as rational or irrational with justification. `(i)sqrt196 , (ii)3sqrt(18), (iii)sqrt(9/27)`
`(iv) sqrt28/sqrt343, (v) -sqrt(0.4), (vi) sqrt12/sqrt75`
(vii) 0.5918, (viii)`(1 + sqrt5)-(4+sqrt5)`
(ix) 10.124124... , 1.010010001...
Thinking process
To classify , use the definiton a rational is in the from of `p/q` where, p and q are integers and `q ne 0` and otherwise it is irrational numbers.

Answer 1

`sqrt196=sqrt(14)^(2)=14`
Hecne, it is a rational number,
` sqrt(9/27)= sqrt(9/(9xx3) =1/(sqrt3)`
Hence, it is an irrational number, because `sqrt3` is an irrational number.
`sqrt28/sqrt343 = sqrt(2xx2xx7)/sqrt(7xx7xx7)= (2sqrt7)/(7sqrt7) = 2/7`
Hence, it is a rational number.
(v) `-sqrt(0.4)=-sqrt(4/10)= -2/sqrt10`
Hence, it is a quotient of rational and irrational number , so it is an irrational number . `sqrt12/sqrt75=sqrt(4xx3)/sqrt(25xx3)= (sqrt4sqrt3)/(sqrt25 sqrt3)=2/5`
Hence, it is a rational number.
(vii) 0.5918, it is a number with terminating decimal, so it can be written in the form of `p/q`
weher `q ne 0.` and q are integers, Hecne, it is a raional number.
(viii) `(1 + sqrt5) - (4 + sqrt5) = 1-4+ sqrt5-sqrt5=-3`
Hence , it is a rational number.
(ix) 10.124124.... , is a number with non- terminating recurring decimal expansion.
Hence, it is a rational number .
(x) 1.010010001........ is a number with non-terminating non - terminating non - recuring decimal expansion
Hence, it is a irrational number.