(1) $\sqrt{3}×(\sqrt{3}+\frac{1}{\sqrt{3}})$
$=\sqrt{3}×\sqrt{3}+\sqrt{3}×\sqrt{\frac{1}{3}}$
$=3 + 1$
$=4$
(2) $\sqrt{18}÷\sqrt{3}+\sqrt{48}×\sqrt{\frac{1}{2}}$
$=\sqrt{6} + \sqrt{48×\frac{1}{2}}$
$=\sqrt{6} + \sqrt{24}$
$=\sqrt{6} + 2\sqrt{6}$
$=3\sqrt{6}$
(3) $(\sqrt{50}+\sqrt{32}-\sqrt{3}×\sqrt{6})÷\sqrt{8}$
$=(5\sqrt{2} + 4\sqrt{2} - \sqrt{18})÷2\sqrt{2}$
$=(9\sqrt{2} - 3\sqrt{2})÷2\sqrt{2}$
$=6\sqrt{2}÷2\sqrt{2}$
$=3$
(4) $(\sqrt{48}-\sqrt{27})÷\sqrt{3}+\sqrt{6}×2\sqrt{\frac{1}{3}}$
$=(4\sqrt{3} - 3\sqrt{3})÷\sqrt{3} + 2\sqrt{6×\frac{1}{3}}$
$=\sqrt{3}÷\sqrt{3} + 2\sqrt{2}$
$=1 + 2\sqrt{2}$

(1) $(\sqrt{7}+\sqrt{3})×(\sqrt{7}-\sqrt{3})-\sqrt{16}$
$=(\sqrt{7})^2 - (\sqrt{3})^2 - 4$
$=7 - 3 - 4$
$=0$
(2) $(2 - \sqrt{5})^2$
$=2^2 - 2×2×\sqrt{5} + (\sqrt{5})^2$
$=4 - 4\sqrt{5} + 5$
$=9 - 4\sqrt{5}$
(3) $(3\sqrt{2} - 1)^2 - (\sqrt{3} + 2)×(\sqrt{3} - 2)$
$=(3\sqrt{2})^2 - 2×3\sqrt{2}×1 + 1^2 - [(\sqrt{3})^2 - 2^2]$
$=18 - 6\sqrt{2} + 1 - (3 - 4)$
$=19 - 6\sqrt{2} - (-1)$
$=20 - 6\sqrt{2}$
(4) $(1 + \sqrt{2} - \sqrt{3})×(1 - \sqrt{2} - \sqrt{3})$
$=[(1 - \sqrt{3}) + \sqrt{2}]×[(1 - \sqrt{3}) - \sqrt{2}]$
$=(1 - \sqrt{3})^2 - (\sqrt{2})^2$
$=1 - 2\sqrt{3} + 3 - 2$
$=2 - 2\sqrt{3}$
(5) $(\sqrt{3} + 1)^2 - (\sqrt{3} + \sqrt{2})×(1 - \sqrt{6})$
$=(\sqrt{3})^2 + 2×\sqrt{3}×1 + 1^2 - [\sqrt{3}×1 + \sqrt{3}×(-\sqrt{6}) + \sqrt{2}×1 + \sqrt{2}×(-\sqrt{6})]$
$=3 + 2\sqrt{3} + 1 - [\sqrt{3} - 3\sqrt{2} + \sqrt{2} - 2\sqrt{3}]$
$=4 + 2\sqrt{3} - (-\sqrt{3} - 2\sqrt{2})$
$=4 + 2\sqrt{3} + \sqrt{3} + 2\sqrt{2}$
$=4 + 3\sqrt{3} + 2\sqrt{2}$
(6) $(2\sqrt{2} + 3)^{2024}(2\sqrt{2} - 3)^{2025} - 4\sqrt{\frac{1}{8}} - \sqrt{(1 - \sqrt{2})^2}$
$=[(2\sqrt{2} + 3)(2\sqrt{2} - 3)]^{2024}(2\sqrt{2} - 3) - 4×\frac{\sqrt{2}}{4} - (\sqrt{2} - 1)$
$=(8 - 9)^{2024}(2\sqrt{2} - 3) - \sqrt{2} - \sqrt{2} + 1$
$=(-1)^{2024}(2\sqrt{2} - 3) - 2\sqrt{2} + 1$
$=1×(2\sqrt{2} - 3) - 2\sqrt{2} + 1$
$=2\sqrt{2} - 3 - 2\sqrt{2} + 1$
$=-2$