Following Bill/Number's denesting rule, I attempted to denest the follow nested radical:
$$\sqrt{2+\sqrt3}$$
The norm evaluates to $4-3=1$, and subtracting it out gives $1+\sqrt3$. The trace evaluates to $\sqrt4=2$, so dividing it out yields $\dfrac{1+\sqrt3}2$; however, this is not equal to the original expression. What is the error I made?
You take the trace of $1+\sqrt3$ not $2+\sqrt3$. You get $$\sqrt{2+\sqrt3}=\frac{1+\sqrt3}{\sqrt2}.$$