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Question Number 216739 by ArshadS last updated on 17/Feb/25

$$Provethat{}^3\sqrt{\sqrt{5}+2}{-}^3\sqrt{\sqrt{5}-2}=1$$$$\text{You can't use 'macro parameter character \#' in math mode}$$

Answered by Rasheed.Sindhi last updated on 17/Feb/25

$$\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}=1$$$$let\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}=x$$$$Anda=\sqrt{5}+2,b=\sqrt{5}-2$$$$\Rightarrow a-b=4,ab=1$$$$a-b={\left(\sqrt[3]{a}\right)}^3-{\left(\sqrt[3]{b}\right)}^3$$$$=(\sqrt[3]{a}-\sqrt[3]{b})({\left(\sqrt[3]{a}\right)}^2+{\left(\sqrt[3]{b}\right)}^2+\left(\sqrt[3]{a}\right)\left(\sqrt[3]{b}\right))$$$$=(\sqrt[3]{a}-\sqrt[3]{b})({(\sqrt[3]{a}-\sqrt[3]{b})}^2+3\left(\sqrt[3]{a}\right)\left(\sqrt[3]{b}\right))$$$$4=\left(x\right)({\left(x\right)}^2+3\left(1\right))$$$$x^3+3x-4=0$$$$(x-1)(x^2+x+4)=0$$$$x=1or{x}^2+x+4=0\Rightarrow x\notin \mathbb{R}$$$$proved$$

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