[tex]\displaystyle\\
a = \frac{1}{2- \sqrt{3}}+ \sqrt{7+4 \sqrt{3} }=\\\\
= \frac{1(2+ \sqrt{3})}{(2- \sqrt{3})(2+ \sqrt{3})}+ \sqrt{4+3+2\times 2\times \sqrt{3} }=\\\\
= \frac{2+ \sqrt{3}}{2^2- (\sqrt{3})^2}+ \sqrt{4+2\times 2\times \sqrt{3}+3 }=\\\\
= \frac{2+ \sqrt{3}}{4- 3}+ \sqrt{2^2+2\times 2\times \sqrt{3}+ (\sqrt{3})^2 }=\\\\
= \frac{2+ \sqrt{3}}{1}+ \sqrt{(2+\sqrt{3})^2 }=(2+ \sqrt{3}) + (2+ \sqrt{3}) = \boxed{2(2+ \sqrt{3})} [/tex]
[tex]\displaystyle\\
b = 2- \sqrt{3}+ \frac{1}{2+ \sqrt{3}} = \\\\
= 2- \sqrt{3}+ \frac{1(2- \sqrt{3})}{(2+ \sqrt{3})(2- \sqrt{3})} =\\\\
= 2- \sqrt{3}+ \frac{2- \sqrt{3}}{2^2- (\sqrt{3})^2} =\\\\
= 2- \sqrt{3}+ \frac{2- \sqrt{3}}{4- 3} =\\\\
=2- \sqrt{3}+ \frac{2- \sqrt{3}}{1} = (2- \sqrt{3})+(2- \sqrt{3})=\boxed{2(2- \sqrt{3})}
[/tex]
[tex]\displaystyle\\
m_g = \sqrt{a\times b} = \sqrt{2(2+ \sqrt{3}) \times 2(2- \sqrt{3})} =\\\\
= \sqrt{2^2 (2+ \sqrt{3})(2- \sqrt{3})} =\sqrt{2^2 \times(2^2- (\sqrt{3})^2)} =\\\\
=\sqrt{4 \times(4- 3)} =\sqrt{4 \times 1} =\sqrt{4} = \boxed{\boxed{2}}[/tex]
