3−32+3=a+b3\frac{3 - \sqrt{3}}{2 + \sqrt{3}} = a + b\sqrt{3}2+33−3=a+b3
Rationalize
= 3−32+3×2−32−3\frac{3 - \sqrt{3}}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}}2+33−3×2−32−3
= (3−3)(2+3)(2−3)(2−3)\frac{(3 - \sqrt{3})}{(2 + \sqrt{3})} \frac{(2 - \sqrt{3})}{(2 - \sqrt{3})}(2+3)(3−3)(2−3)(2−3)
= 6−33−23+(3)24−23+23−(3)2\frac{6 - 3\sqrt{3} - 2\sqrt{3} + (\sqrt{3})^2}{4 - 2\sqrt{3} + 2\sqrt{3} - (\sqrt{3})^2}4−23+23−(3)26−33−23+(3)2
= 6−53+34−3\frac{6 - 5\sqrt{3} + 3}{4 - 3}4−36−53+3
= 9−531=9−53\frac{9 - 5\sqrt{3}}{1} = 9 - 5\sqrt{3}19−53=9−53
= 9 + (-5) 3\sqrt{3}3
∴a=9,b=−5\therefore a = 9, b = -5∴a=9,b=−5