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Q.
Let $a$ and $b$ be two rational numbers. If $a b$ is a rational number where $a=2-\sqrt{3}$. Find the value of $b$.
Real Numbers
Solution:
Going by options,
$a=2-\sqrt{3}, b=2+\sqrt{3}$
(i) $a b=(2-\sqrt{3})(2+\sqrt{3})$
$=(2)^2-(\sqrt{3})^2$
$=4-3=1$ is a rational number
(ii) $a=2-\sqrt{3}, b=2(2+\sqrt{3})$
$\Rightarrow a b=(2-\sqrt{3}) 2(2+\sqrt{3}) $
$=2(1) $
$=2 \text { is a rational number }$
(iii) $a=2-\sqrt{3}$ and $b=4(2+\sqrt{3})$
$a b=(2-\sqrt{3}) 4(2+\sqrt{3}) $
$=4(1) $
$=4 \text { is a rational number }$