Positive rational numbers $a$,$b$ satisfy $a^3+4a^2b=4a^2+b^4$. Prove that $\sqrt{\sqrt{a}-1}$ is also rational. 
Positive rational numbers $a$,$b$ satisfy $a^3+4a^2b=4a^2+b^4$. Prove that $\sqrt{\sqrt{a}-1}$ is also rational.

My try: $a(a+2b)^2=a^3+4a^2b+4b^2a=4a^2+b^4+4b^2a=(2a+b^2)^2$, so $\sqrt{a}=\frac{2a+b^2}{a+2b}$.

Your approach is correct you can just continue, by proving that
$$\sqrt a-1=\left(\frac b{\sqrt a}-1\right)^2$$
and remember that $\sqrt a$ is proved to be rational!
