Question

If $\sqrt{a},\sqrt{b}$ are unlike surds, then show that there exist no surd $\sqrt{c}$ satisfying $\sqrt{a}+\sqrt{b}=\sqrt{c}$

Answer 1

$\sqrt{a}+\sqrt{b}=\sqrt{c}$ (Assume true)

square both sides

$a+b+2\sqrt{ab}=c$

As $\sqrt{a}\sqrt{b}$ are unlike surds $\therefore$

$2\sqrt{ab}$ is irrational $(c-a-b)$ is rational $\therefore$ wrong assumption.