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$\sqrt{17+2\sqrt{30} } =\sqrt{(15+2)+2\sqrt{30} } =\sqrt{15+2\sqrt{15*2}+2 } =\\ \\ \sqrt{(\sqrt{15)} ^{2}+2\sqrt{15*2}+(\sqrt{2} )^{2} } =\sqrt{(\sqrt{15} +\sqrt{2} )^{2} } =\\ \\ \sqrt{15} +\sqrt{2}$

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Объяснение:

$\bf\\\sqrt{17+2\sqrt{30} } =\sqrt{2+2\sqrt{2\cdot15} +15}=\sqrt{(\sqrt{2} )^2+2\sqrt{2}\cdot\sqrt{15}+(\sqrt{15} )^2 } =\\\\\\=\sqrt{(\sqrt{2}+\sqrt{15})^2 }=|\sqrt{2}+\sqrt{15}|=\sqrt{2}+\sqrt{15}$

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