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Question Number 43349 by peter frank last updated on 10/Sep/18

Answered by math1967 last updated on 10/Sep/18

$$\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{(3-\sqrt{{5)}^2}}}}$$$$\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+3-\sqrt{5}}}$$$$\sqrt{2+\sqrt{5}-\sqrt{{(\sqrt{5}-2)}^2}}$$$$\sqrt{2+\sqrt{5}-\sqrt{5}+2}$$$$\sqrt{4}=2\left(proved\right)$$

Commented by peter frank last updated on 10/Sep/18

$$muchrespect$$

Answered by Joel578 last updated on 10/Sep/18

$$\sqrt{14-6\sqrt{5}}=\sqrt{14-2\sqrt{45}}=3-\sqrt{5}$$$$$$$$\sqrt{6-3\sqrt{5}+3-\sqrt{5}}=\sqrt{9-4\sqrt{5}}=\sqrt{9-2\sqrt{20}}$$$$=\sqrt{5}-2$$$$$$$$\sqrt{2+\sqrt{5}-\sqrt{5}+2}=\sqrt{4}=2$$

Commented by malwaan last updated on 10/Sep/18

$$3-\sqrt{5}??$$

Commented by Joel578 last updated on 10/Sep/18

$${(3-\sqrt{5})}^2=9-6\sqrt{5}+5=14-6\sqrt{5}=14-2\sqrt{45}$$$$\sqrt{{(3-\sqrt{5})}^2}=\sqrt{14-2\sqrt{45}}$$

Commented by peter frank last updated on 10/Sep/18

$$$$$$$$$$thanks$$

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