<s>[INST] Are there nice ways to solve $(2+x)^{0.25}-(2-x)^{0.25}=1$ Are there nice and elegant ways to solve this equation?
$(2+x)^{0.25}-(2-x)^{0.25}=1$
Thanks.
 [/INST] Let $A=\sqrt{2+x}$ and $B=\sqrt{2-x}$.  See if you can show the following two equations hold:
$$A^2+B^2=4$$
$$2A+2AB+2B=5$$
Adding these two equations together, and using the quadratic formula you should be able to show that $A+B=\sqrt{10}-1$.  Then, plugging this into the second equation, and using the fact that $AB=\sqrt{4-x^2}$, you will obtain the equation
$$7-2\sqrt{10}=2\sqrt{4-x^2}$$
Solving for $x$ gives
$$x=\frac{1}{2}\sqrt{28\sqrt{10}-73}$$
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