Question

prove that √P+√Q is irrational if √PQ is irrational.
​

Answer 1

Answer:

Solution: Let us suppose that √p + √q is rational. ... => √q = a – √pSquaring on both sides we getq = a2 + p - 2a√p=> √p = a2 + p - q/2a which is a contradiction as the right hand side is rational number while √p is irrational. Hence √p + √q is irrational.

Answer 2

Given :-

$\sqrt{PQ}$  is irrational.

Require to prove :-

$\sqrt{P}+ \sqrt{Q}$ is irrational.

Procedure :-

Let us square $\sqrt{P}+ \sqrt{Q}$

$(\sqrt{P}+ \sqrt{Q})^2=P+Q+\sqrt{PQ}$

If the part of the number is irrational, the entire number is irrational.

If we find the root of an irrational number, we definitely get irrational number.

∴ $\sqrt{P}+ \sqrt{Q}$ is irrational.