[tex]\begin{aligned}x*y &=xy-\sqrt{3}(x+y)+3\sqrt{3} \\ &=(x-\sqrt 3)(y-\sqrt 3)+\sqrt{3} \\ &=(y-\sqrt{3})(x-\sqrt{3})+\sqrt{3} \\ &=y*x\to \text{legea este comutativa}\end{aligned}[/tex]
[tex]\\\dfrac{\sqrt{5}}{\sqrt{1}}*\dfrac{\sqrt{6}}{\sqrt{2}}*\dfrac{\sqrt{7}}{\sqrt{3}}*...*\dfrac{\sqrt{100}}{\sqrt{96}} =\\ \\ =\dfrac{\sqrt{5}}{\sqrt{1}}*\sqrt{3}*\dfrac{\sqrt{7}}{\sqrt{3}}*...*\dfrac{\sqrt{100}}{\sqrt{96}}\\ \\ = \sqrt{3}*\underset{a}{\underbrace{\left(\dfrac{\sqrt{5}}{\sqrt{1}}*\dfrac{\sqrt{7}}{\sqrt{3}}*\dfrac{\sqrt{9}}{\sqrt{5}}*...*\dfrac{\sqrt{100}}{\sqrt{96}}\right)}}[/tex]
[tex]=\sqrt{3}*a \\ \\= (\sqrt{3}-\sqrt{3})(a+\sqrt{3})+\sqrt{3}\\ \\ = 0\cdot (a-\sqrt{3})+\sqrt{3}\\ \\= 0+\sqrt{3}\\\\=\boxed{\sqrt{3}}[/tex]
