If x 1/4 + x -1/4 = 2, then what is the value of x 81 + (1/ x 81 )?
2
Let \(y = x^{1/4}\). The equation becomes \(y + \dfrac{1}{y} = 2\), i.e. \(y^2-2y+1 = 0\), so \((y-1)^2 = 0 \implies y = 1\).
Hence \(x^{1/4} = 1 \implies x = 1\).
Therefore \(x^{81} + \dfrac{1}{x^{81}} = 1 + 1 = \mathbf{2}\).
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