Writing complex numbers in form $a+bi$ Can $\sqrt{i+\sqrt{2}}$ be expressed as $a+bi$ with $a,b \in \mathbb{R}$? In general, what kinds of expressions can be rewritten in that form?

You can express such an expression in the form $a+ib$. 
Let $$x+iy = \sqrt{i+\sqrt{2}} \\
      (x+iy)^2 = (\sqrt{i+\sqrt{2}})^2 \\
       x^2 - y^2 +2ixy = i+\sqrt{2} \\
       $$
Now you only need to solve the equations $x^2 - y^2 = \sqrt{2}$, and $xy = \frac{1}{2}$ to get the values of $x$ and $y$. 

 $x = \,\,^+_-\sqrt{\frac{\sqrt{3}+\sqrt{2}}{2}}$ and $y = \,\,^+_-\sqrt{\frac{\sqrt{3}-\sqrt{2}}{2}}$

