Question

Which of the following expressions on simplification gives a rational number .

• A. $\sqrt{7}(2+\sqrt{7})$
• B. $\sqrt{3}(2+\sqrt{3})$
• C. $\sqrt{2}(\sqrt{8}+\sqrt{2})$
• D. None of the above

Answer 1

Answer: (C)

$\sqrt{2}(\sqrt{8}+\sqrt{2})$

Using distributive property $a.(b+c)=a.b+a.c$
For option $A$
$\sqrt{7}(2+\sqrt{7})=2.\sqrt{7}+\sqrt{7}.\sqrt{7}=2\sqrt{7}+7$
which is irrational
$\sqrt{3}(2+\sqrt{3})=2.\sqrt{3}+\sqrt{3}.\sqrt{3}=2\sqrt{3}+3$
which is also irrational
$\sqrt{2}(\sqrt{8}+\sqrt{2})=\sqrt{2}(2\sqrt{2}+\sqrt{2})=2\sqrt{2}.\sqrt{2}+\sqrt{2}.\sqrt{2}=4+2=6$
which is rational
So option $C$ is correct.