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Question Number 217985 by hardmath last updated on 24/Mar/25

$${\mathrm{x}}^3-3\mathrm{x}+1=0$$$$\mathrm{The}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{equation}\to \mathrm{a},\mathrm{b},\mathrm{c}$$$$\mathrm{Find}\to \sqrt[3]{\mathrm{a}}+\sqrt[3]{\mathrm{b}}+\sqrt[3]{\mathrm{c}}=?$$

Commented by Frix last updated on 24/Mar/25

$$\mathrm{suppose}a<b<c$$$$\Rightarrow$$$$a=-2\cos \frac{\pi }{9}$$$$b=2\sin \frac{\pi }{18}$$$$c=2\cos \frac{2\pi }{9}$$$$\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{think}$$$$-\sqrt[3]{2\cos \frac{\pi }{9}}+\sqrt[3]{2\sin \frac{\pi }{18}}+\sqrt[3]{2\cos \frac{2\pi }{9}}$$$$\mathrm{is}\mathrm{any}‘‘{\mathrm{nice}}^{″}\mathrm{number}.$$$$\mathrm{Do}\mathrm{you}\mathrm{have}\mathrm{an}\mathrm{exact}\mathrm{answer}?$$

Commented by hardmath last updated on 24/Mar/25

$$$$Dear professor, I don't know the answer

Commented by Ghisom last updated on 26/Mar/25

$$\mathrm{I}\mathrm{found}\mathrm{a}\mathrm{way}\mathrm{to}\mathrm{solve}\mathrm{it}...$$$$a^{1/3}+b^{1/3}+c^{1/3}={(3^{5/3}-6)}^{1/3}$$

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