How to simplify $(a^2+ab+b^2)/(a+\sqrt{ab}+b)$ How can I simplify as much as possible:
$$\frac{a^2+ab+b^2}{a+\sqrt{ab}+b}$$
Also, first post here, looking forward to sticking around!

You could multiply by the conjugate of the denominator, then cancel.
\begin{align*}
\frac{a^2 + ab + b^2}{a + \sqrt{ab} + b} & = \frac{a^2 + ab + b^2}{a + b + \sqrt{ab}} \cdot \frac{a + b - \sqrt{ab}}{a + b - \sqrt{ab}}\\
& = \frac{(a^2 + ab + b^2)(a + b - \sqrt{ab})}{(a + b)^2 - ab}\\
& = \frac{(a^2 + ab + b^2)(a + b - \sqrt{ab})}{a^2 + 2ab + b^2 - ab}\\
& = \frac{(a^2 + ab + b^2)(a + b - \sqrt{ab})}{a^2 + ab + b^2}\\
& = a + b - \sqrt{ab}
\end{align*}
