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$\frac{\sqrt{57+40\sqrt2}-\sqrt{57-40\sqrt2}}{2}=\frac{\sqrt{(\sqrt{32}+\sqrt{25})^2}-\sqrt{(\sqrt{32}-\sqrt{25})^2}}{2}=\\\\= \frac{|\sqrt{32}+\sqrt{25}|-|\sqrt{32}-\sqrt{25}|}{2} = \frac{\sqrt{32}+\sqrt{25}-(\sqrt{32}-\sqrt{25})}{2}=\\\\= \frac{\sqrt{25}+\sqrt{25}}{2} =\frac{5+5}{2}=5$

$\star \; \; \sqrt{57+40\sqrt2} =\sqrt{32+25+2\cdot \sqrt{20^2\cdot 2}}=\\\\=\sqrt{32+25+2\cdot \sqrt{800}}=\sqrt{32+25+2\cdot \sqrt{{32\cdot 25}}}=\\\\=\sqrt{(\sqrt{32}+\sqrt{25})^2}=|\sqrt{32}+\sqrt{25}|=\sqrt{32}+\sqrt{25}=$

$=\sqrt{2^5}+5=\sqrt{2^4\cdot 2}+5=2^2\cdot \sqrt2+5=4\sqrt2+5\; ;\\\\\sqrt{57-20\sqrt2}=\sqrt{32}-\sqrt{25}=4\sqrt2-5}\; .$