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Question Number 10193 by konen last updated on 29/Jan/17

$$\mathrm{x}{=}^3\sqrt{11+\sqrt{57}}+{}^3\sqrt{11-\sqrt{57}}$$$$\Rightarrow {\mathrm{x}}^3-12\mathrm{x}=?$$

Answered by ridwan balatif last updated on 29/Jan/17

$${\mathrm{x}}^3={({}^3\sqrt{11+\sqrt{57}}+{}^3\sqrt{11-\sqrt{57}})}^3$$$${\mathrm{x}}^3={\left({}^3\sqrt{11+\sqrt{57}}\right)}^3+{\left({}^3\sqrt{11-\sqrt{57}}\right)}^3+3\times \left({}^3\sqrt{11+\sqrt{57}}\right)\left({}^3\sqrt{11-\sqrt{57}}\right)\left({}^3\sqrt{11+\sqrt{57}}{+}^3\sqrt{11-\sqrt{57}}\right)$$$$\mathrm{remember}:{(\mathrm{a}+\mathrm{b})}^3={\mathrm{a}}^3+{\mathrm{b}}^3+3\mathrm{ab}(\mathrm{a}+\mathrm{b})$$$$\mathrm{and}\mathrm{also}\mathrm{remember}\mathrm{x}{=}^3\sqrt{11+\sqrt{57}}{+}^3\sqrt{11-\sqrt{57}},\mathrm{so}$$$${\mathrm{x}}^3=11+\sqrt{57}+11-\sqrt{57}+3\times \left({}^3\sqrt{121-57}\right)\left(\mathrm{x}\right)$$$${\mathrm{x}}^3=22+3\mathrm{x}.\left({}^3\sqrt{64}\right)$$$${\mathrm{x}}^3=22+12\mathrm{x}$$$${\mathrm{x}}^3-12\mathrm{x}=22$$

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