Question

Let $a$ and $b$ be two rational numbers. If $ab$ is a rational number where $a=2-\sqrt{3}$. Find the value of $b$.

• A. $2+\sqrt{3}$
• B. $2(2+\sqrt{3})$
• C. $4(2+\sqrt{3})$
• D. All of these

Answer 1

Answer: (D)

Going by options,
$a=2-\sqrt{3},b=2+\sqrt{3}$
(i) $ab=(2-\sqrt{3})(2+\sqrt{3})$
$=(2{)}^2-(\sqrt{3}{)}^2$
$=4-3=1$ is a rational number
(ii) $a=2-\sqrt{3},b=2(2+\sqrt{3})$
$\Rightarrow ab=(2-\sqrt{3})2(2+\sqrt{3})$
$=2(1)$
$=2\text{ is a rational number }$
(iii) $a=2-\sqrt{3}$ and $b=4(2+\sqrt{3})$
$ab=(2-\sqrt{3})4(2+\sqrt{3})$
$=4(1)$
$=4\text{ is a rational number }$