6.23. Find the first digit after the decimal point of the number $(2+\sqrt{3})^{1000}$.

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6.23. Answer: 9. Let $(2+\sqrt{3})^{n}=A_{n}+B_{n} \sqrt{3}$, where $A_{n}$ and $B_{n}$ are natural numbers. According to problem 6.22, $(2-\sqrt{3})^{n}=A_{n}-B_{n} \sqrt{3}$. Therefore, $(2+\sqrt{3})^{n}+(2-\sqrt{3})^{n}$ is a natural number. But $2-\sqrt{3} \approx$ $\approx 0.2679 < 0.3$, so $(2-\sqrt{3})^{1000} \approx 0.0 \ldots$ (and then several more zeros follow).