Question

prove p√q is irrational

Answer 1

let p√q be rational
p√q=a/b where a,b are co prime integers
√q=a/pb
irrational not equals to rational
it contradicts the fact
so, p√q is irrational

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Answer 2

Let
see
√p+√q is rational number

A rational number can be written in the form of a/b

√p+√q=a/b
√p=a/b-√q
√p=(a-b√q)/b

p, q are integers then (a-b√q) /b is a rational number.

So, √p is also a rational number.

But this contradicts the fact that √p is an irrational number.

so, our supposition is false

√p+√q is an irrational number..

hope helped
@srk6