Question:
$(-2-\sqrt{3})(-2+\sqrt{3})$ when simplified is
(a) positive and irrational
(b) positive and rational
(c) negative and irrational
(d) negative and rational
Solution:
$(-2-\sqrt{3})(-2+\sqrt{3})$
$=-(2+\sqrt{3}) \times[-(2-\sqrt{3})]$
$=(2+\sqrt{3})(2-\sqrt{3})$
$=2^{2}-(\sqrt{3})^{2} \quad\left[(a+b)(a-b)=a^{2}-b^{2}\right]$
$=4-3$
$=1$
Thus, the given expression when simplified is positive and rational.
Hence, the correct answer is option (b).