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Question Number 185354 by Ml last updated on 20/Jan/23

Answered by HeferH last updated on 20/Jan/23

$$\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}=k$$$$2+\sqrt{3}+2-\sqrt{3}-2(\sqrt{2+\sqrt{3}}\cdot \sqrt{2-\sqrt{3}})=k^2$$$$4-2\left(\sqrt{2^2-{\left(\sqrt{3}\right)}^2}\right)=k^2$$$$\sqrt{2}=k$$

Answered by mr W last updated on 20/Jan/23

$$2\pm \sqrt{3}=\frac{4\pm 2\sqrt{3}}{2}=\frac{{(\sqrt{3}\pm 1)}^2}{2}$$$$\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}=\frac{\sqrt{3}+1}{\sqrt{2}}-\frac{\sqrt{3}-1}{\sqrt{2}}=\sqrt{2}$$

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