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Question Number 139165 by mathlove last updated on 23/Apr/21

Answered by mr W last updated on 23/Apr/21

$$13+\sqrt{48}={\left(2\sqrt{3}\right)}^2+2\times 2\sqrt{3}+1={(2\sqrt{3}+1)}^2$$$$5-\sqrt{13+\sqrt{48}}=5-(2\sqrt{3}+1)={\left(\sqrt{3}\right)}^2-2\sqrt{3}+1={(\sqrt{3}-1)}^2$$$$3+\sqrt{5-\sqrt{13+\sqrt{48}}}=3+(\sqrt{3}-1)=2+\sqrt{3}$$$$\sqrt{2-\sqrt{3}}\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}=\sqrt{2-\sqrt{3}}\sqrt{2+\sqrt{3}}$$$$=\sqrt{2^2-{\left(\sqrt{3}\right)}^2}=\sqrt{4-3}=1$$

Commented by mathlove last updated on 23/Apr/21

$$thankssir$$

Commented by greg_ed last updated on 23/Apr/21

$$\mathrm{cool}!$$

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