Question

prove that underroot 2,3,5 are irrational no.'s (please with easy language and step by step)

Answer 1

see now underroot 2,3,5 are the irrational number because it is not terminating non recurring..
the meaning of non terminating non recurring is the number which is infinite and in non type

for example 2.39274619329372187..........

now the underroot of 2 is 1.414........,...183,,,......... etc
now the underroot of 3 is 1.730...........18339297.....
now the underroot of 5 is 2.236...........


for example now the underroot of 9 is 3 which is terminating number....

ok

thank you

Answer 2

let us assume on the contrary that root 2 is a rational. then ,there exist positive integers p and q such that
root 2 =a/b where p and q ,are co-prime I.e.their HCF is 1
root 2= a/b(squaring both side )
2 =a/b(square on p/q )
2 |a square [therefore 2|2b square and 2 b square=a square
a=2c[ the fundamental theorem ]
a square= 4 c square
2b square =4c square [therefore 2b square =a square
b square =2 c square [therefore 2 |2 c square
2| b square
2|b
we obtain that 2 is a common factor of à and b .but ,this contradicts the fact that a and b have no common factor other than 1 . this means that our supposition is wrong.
hence,root 2 is an irrational number
I hope it help you!!!