Question

If a, b, c are rational such that $a+b\sqrt{2}+c\sqrt{3}=0$, then:

• A. $b=c=0$
• B. $a=0$
• C. $a=b=c=0$
• D. $ab=1$

Answer 1

Answer: (C)

$a=b=c=0$

Given that $a,b,c$ are rational numbers.

Also,

$a+b\sqrt{2}+c\sqrt{3}=0$

i.e. $RHS=0$ which is a rational number.

Then $\Rightarrow$ $LHS=a+b\sqrt{2}+c\sqrt{3}$ must also be rational.

We know that $a,b,c$ are rational. Also, addition and multiplication of a rational number with an irrational number gives irrational number.

$\therefore$ Only possible combination is $a=b=c=0$.

Therefore, option $C$ is correct.