(1) \( x^{2}+\frac{1}{x^{2}} \)

Did you know that the expression \( x^{2} + \frac{1}{x^{2}} \) can be linked to geometric interpretations? If you consider a circle with a radius \( r = x \) and \( r = \frac{1}{x} \), this expression can represent the sum of the squares of two lengths derived from those radii. Another fun aspect is how you can manipulate this expression if you're given \( x + \frac{1}{x} \). In fact, if you know that value, you can derive \( x^{2} + \frac{1}{x^{2}} \) quite easily using the identity \((x + \frac{1}{x})^{2} = x^{2} + 2 + \frac{1}{x^{2}}\). Just rearrange to find \( x^{2} + \frac{1}{x^{2}} = (x + \frac{1}{x})^{2} - 2 \)!