Question

If $a,b,c,d$ are rationals $b>0,d>0$ and $\sqrt{b},\sqrt{d}$ are surds and $a+\sqrt{b}=c+\sqrt{d}$, then show that $a=c,b=d$

Answer 1

Given, $a+\sqrt{b}=c+\sqrt{d}$, where $a$ & $c$ are rational &

As $\sqrt{b}$ & $\sqrt{d}$ are irrational,

$\therefore$ the sum of rational & irrational is equal only if the individual numbers are equal,

i.e. $a=c$ & $b=d$.

Therefore, hence showed.