[tex]\displaystyle\\x=\Big|-2\sqrt{5}-4\Big|=\Big|-\Big(2\sqrt{5}+4\Big)\Big|=\boxed{\bf2\sqrt{5}+4}\\\\y=\sqrt{\Big(4-2\sqrt{5}\Big)^\b2}}=\Big|4-2\sqrt{5}\Big|\\4^2=16~~~si~~~\Big(2\sqrt{5}\Big)^{\b2}=4\cdot5=20\\\implies~~2\sqrt{5}>4\\\\\implies~~y=\sqrt{\Big(4-2\sqrt{5}\Big)^\b2}}=\Big|4-2\sqrt{5}\Big|=\boxed{\bf2\sqrt{5}-4}[/tex]
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[tex]\displaystyle\\m_g=\sqrt{x\cdot y}=\sqrt{\Big(2\sqrt{5}+4\Big)\cdot\Big(2\sqrt{5}-4\Big)}=\\\\=\sqrt{\Big(2\sqrt{5}\Big)^{\b2}-\Big(4\Big)^{\b2}}=\sqrt{4\cdot5-16}=\sqrt{20-16}=\sqrt{4}=\boxed{\bf2}[/tex]
