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Algebra Difficulty 5.0 AIME, harder Find the answer Ukraine

Find the integer that is closest to the value of the expression:
((3+1)2023−(13−1)2023)⋅((3+2)2023−(13−2)2023)⋅((3+3)2023−(13−3)2023)…((3+8)2023−(13−8)2023). \left( (3 + \sqrt{1})^{2023} - \left( \frac{1}{3 - \sqrt{1}} \right)^{2023} \right) \cdot \left( (3 + \sqrt{2})^{2023} - \left( \frac{1}{3 - \sqrt{2}} \right)^{2023} \right) \cdot \left( (3 + \sqrt{3})^{2023} - \left( \frac{1}{3 - \sqrt{3}} \right)^{2023} \right) \dots \left( (3 + \sqrt{8})^{2023} - \left( \frac{1}{3 - \sqrt{8}} \right)^{2023} \right).

Solution

Let's consider the last factor:
13−8=3+89−8=3+8⇒(3+8)2023=(13−8)2023, \frac{1}{3 - \sqrt{8}} = \frac{3 + \sqrt{8}}{9 - 8} = 3 + \sqrt{8} \Rightarrow (3 + \sqrt{8})^{2023} = \left(\frac{1}{3 - \sqrt{8}}\right)^{2023},
which means that the last factor equals 00, and therefore the whole product equals 00.

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