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Question Number 116987 by ZiYangLee last updated on 08/Oct/20

$$\mathrm{Find}\mathrm{the}\mathrm{largest}\mathrm{integer}\mathrm{smaller}\mathrm{than}$$$${(7+4\sqrt{3})}^3.$$

Answered by 1549442205PVT last updated on 08/Oct/20

$$\mathrm{a}={(7+4\sqrt{3})}^3=343+588\sqrt{3}+1008+64\sqrt{3}$$$$\mathrm{b}={(7-4\sqrt{3})}^3=343-588\sqrt{3}+1008-64\sqrt{3}$$$$\mathrm{a}+\mathrm{b}=1341\times 2=2682$$$$0<\mathrm{b}={(7-4\sqrt{3})}^3<1\Rightarrow \left[(7+4\sqrt{3})\right]=2681$$

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