If $x+\frac{1}{x}=6$, then what is the value of $x^{2}+\frac{1}{x^{2}}$?
This problem looks like it involves some algebra and manipulation of fractions. I wonder what would happen if I square both sides of the equation. Squaring the left side, I get $(x+\frac{1}{x})^{2}=x^{2}+2x\cdot\frac{1}{x}+\frac{1}{x^{2}}$. Simplifying, I get $x^{2}+2+\frac{1}{x^{2}}$. Squaring the right side, I get $6^{2}=36$. So I have the equation $x^{2}+2+\frac{1}{x^{2}}=36$. I want to isolate $x^{2}+\frac{1}{x^{2}}$, so I subtract 2 from both sides. I get $x^{2}+\frac{1}{x^{2}}=34$. 