To prove :- root p + root q is an irrational number Since in the given proof , - Maths -

Question

To prove :- root p + root q is an irrational number
Since in the given proof , we proved that root q is rational , then
how root p + root q can be irrtaional
Please explain !! Thank you experts

Laksha Leap · Answer

(This proof only valids when p and q both are non-perfect square
number )Let root p + root q is rational Consider 2 co-prime
integers a and b such that root p + root q = a/b =》a/b -
root q = root p =》(a/b - root q)^2 = root p^2 = p
=》(a^2/b^2 + q - 2a×rootq/b ) = p =》(a^2/b^2 +
q - p) × b/2a = root q =》root q = (a^2/b^2 + q - p)
× b/2a Now, root q is irrational as q is non-perfect square
number, but RHS value of root q above is rational This is a
contradiction which has arisen by assuming root p + root q is
rational This contradiction shows the irrationality of root p +root
q (If it would be given that either of p and q is perfect square,
then, either of root q and root q will be rational then, In root p
+ root q , one term (root p or root q) will be irrational and other
will be rational Then, we can easily prove its irrationality as the
sum of a rational and irrational is always irational )

To prove :- root p + root q is an irrational number . Since in the given proof , we proved that root q is rational , then how root p + root q can be irrtaional . Please explain ... !! Thank you experts .

To prove :- root p + root q is an irrational number .
Since in the given proof , we proved that root q is rational , then how root p + root q can be irrtaional .
Please explain ... !! Thank you experts .