◆ NUMBER SYSTEMS ◆
[tex] \frac{7 - 3 \sqrt{3} }{2 + \sqrt{3} } \\ \\ rationalising \: the \: denominator \: , \\ \\ \frac{7 - 3 \sqrt{3} }{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ = \frac{(7 - 3 \sqrt{3})(2 - \sqrt{3}) }{ {2}^{2} - { \sqrt{3} }^{2} } \\ \\ = \frac{14 - 7 \sqrt{3} - 6 \sqrt{3} + 9 }{4 - 3} \\ \\ = \frac{23 - 13 \sqrt{3} }{1} \\ \\ = 23 - 13 \sqrt{3} = a + b \sqrt{3} \\ \\ comparing \: both \: sides \: , \\ we \: get \: , \\ \\ a = 23 \\ and \: , \: \\ b = - 13[/tex]
