If $$x = \frac{\sqrt{101} + 1}{2}$$, then find the value of $$(x^3 - 26x - 23)^5$$

Solution

From $$2x - 1 = \sqrt{101}$$ we get $$4x^2 - 4x + 1 = 101$$, so $$x^2 = x + 25$$. Then $$x^3 = x \cdot x^2 = x^2 + 25x = 26x + 25$$, so $$x^3 - 26x - 23 = 25 - 23 = 2$$. Therefore the required value is $$2^5 = 32$$.

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