Verified answer

[tex]\displaystyle\bf\\1)\\\\\sqrt{\Big(7-\sqrt{2}\Big )^{2} } +\sqrt{\Big(1-\sqrt{2}\Big )^{2} } =\Big|7-\sqrt{2} \Big|+\Big|1-\sqrt{2} \Big|=\\\\\\=7-\sqrt{2} +\sqrt{2} -1=6\\\\\\2)\\\\\Big(5+3\sqrt{7} \Big)\cdot\Big(2-\sqrt{3} \Big)^{2} =\Big(5+3\sqrt{7} \Big)\cdot\Big(4-4\sqrt{3} +3\Big)=\\\\\\=\Big(5+3\sqrt{7} \Big)\cdot\Big(7-4\sqrt{3} \Big)=35-20\sqrt{3} +21\sqrt{3} -12\sqrt{21} =\\\\\\=35+\sqrt{3}-12\sqrt{21}[/tex]