<s>[INST] Prove that $\frac{a-x}{\sqrt{a}-\sqrt{x}}-\left(\frac{a+\sqrt[4]{ax^3}}{\sqrt{a}+\sqrt[4]{ax}}-\sqrt[4]{ax}\right)=2\sqrt[4]{ax}$ Prove that $$\dfrac{a-x}{\sqrt{a}-\sqrt{x}}-\left(\dfrac{a+\sqrt[4]{ax^3}}{\sqrt{a}+\sqrt[4]{ax}}-\sqrt[4]{ax}\right)=2\sqrt[4]{ax}, a>0, x>0,x\ne a.$$ My try $$\dfrac{a-x}{\sqrt{a}-\sqrt{x}}-\left(\dfrac{a+\sqrt[4]{ax^3}}{\sqrt{a}+\sqrt[4]{ax}}-\sqrt[4]{ax}\right)=\dfrac{\sqrt{a}^2-\sqrt{x}^2}{\sqrt{a}-\sqrt{x}}-\dfrac{a+\sqrt[4]{ax^3}-\sqrt[4]{a^3x}-\sqrt[4]{a^2x^2}}{\sqrt{a}+\sqrt[4]{ax}}$$ Can we do something more directly to arrive at the solution? Thank you!
 [/INST] Just to make things easier to type let $c =\sqrt[4]a$ and $y=\sqrt[4]x$ so we are asked to prove
$\frac {c^4 - y^4}{c^2-y^2} -(\frac {c^4 + cy^3}{c^2 + cy} - cy)= 2cy$.
so lets do that
$\frac {c^4 - y^4}{c^2-y^2} -(\frac {c^4 + cy^3}{c^2 + cy} - cy)=$
$\frac {(c^2-y^2)(c^2+y^2)}{c^2-y^2} -\frac {c(c^3+ y^3)}{c(c+y)} + cy=$
$c^2+cy+y^2 -\frac {c^3+y^3}{c+y}=$
$c^2 + cy + y^2 - \frac {(c+y)(c^2 -cy+y^2)}{c+y}=$
$c^2 + cy + y^2 -(c^2 -cy +y^2) = 2cy$
That's it.
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