How to simplify $(a^2+ab+b^2)/(a+\sqrt{ab}+b)$

Question

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!

Answer 1

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*}

Answer 2

HINT:

$$x^4+x^2y^2+y^4=(x^2+y^2)^2-(xy)^2$$

Answer 3

$$\frac{a^2 + ab + b^2}{a + \sqrt{ab} + b}= \frac{(a^3-b^3)/(a-b)}{(a^{3/2} -b^{3/2})/(a^{1/2} - b^{1/2}) } =\frac{a^{3/2} + b^{3/2}}{a^{1/2} + b^{1/2}}=a-\sqrt{ab} + b$$