<s>[INST] Evaluating $x^4 + \frac{1}{x^4}$ given that $x^2 - 3x + 1 = 0$ Determine the value of $x^4 + \frac{1}{x^4}$ given that $x^2 - 3x + 1 = 0$. I've tried forcing in a difference of squares, looked for various difference of $n$s or sum of odd powers that I could equate this to, but have yet to find a solution. 
 [/INST] Hint Dividing gives $$x - 3 + \frac{1}{x} = 0,$$ and then rearranging gives
$$x + \frac{1}{x} = 3.$$

Additional hint Squaring gives $$3^2 = \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2},$$ and rearranging again gives $$x^2 + \frac{1}{x^2} = 7.$$

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