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Question Number 119848 by benjo_mathlover last updated on 27/Oct/20

$$Solveinrealnumberstheequation$$$$\sqrt[3]{x}+\sqrt[3]{x-1}+\sqrt[3]{x+1}=0$$

Answered by 1549442205PVT last updated on 27/Oct/20

$$\mathrm{Applying}\mathrm{the}\mathrm{identity}$$$${\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=3\mathrm{abc}\mathrm{when}\mathrm{a}+\mathrm{b}+\mathrm{c}=0.\mathrm{Then}$$$$\sqrt[3]{x}+\sqrt[3]{x-1}+\sqrt[3]{x+1}=0\mathrm{gives}\mathrm{us}$$$$3\mathrm{x}=3{}^3\sqrt{\mathrm{x}({\mathrm{x}}^2-1)}\iff {\mathrm{x}}^3=\mathrm{x}({\mathrm{x}}^2-1)$$$$\iff \mathrm{x}=0.\mathrm{Thus}\mathrm{x}=0\mathrm{is}\mathrm{unique}\mathrm{root}\mathrm{of}$$$$\mathrm{given}\mathrm{equation}$$$$$$

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