Question

State whether the given statement is true/false:

$\sqrt{p}+\sqrt{q}$, is irrational, where p,q are primes.

• A. True
• B. False

Answer 1

The correct option is A True

Assume that $\sqrt{p}+\sqrt{q}$ is rational. So,

$\sqrt{p}+\sqrt{q}=\frac{a}{b}$

$p+q+2\sqrt{pq}=\frac{a^2}{b^2}$

$\sqrt{pq}=\frac{1}{2}\left(\frac{a^2}{b^2}–p-q\right)$

Since the RHS of the above equation is rational but $\sqrt{pq}$ is an irrational number, so the assumption is wrong.

Therefore, it is true that $\sqrt{p}+\sqrt{q}$ is irrational.