<s>[INST] Show that $\sqrt[3]{3\sqrt{21} + 8} - \sqrt[3]{3\sqrt{21} - 8} = 1$ 
Show that
  $$\sqrt[3]{3\sqrt{21} + 8} - \sqrt[3]{3\sqrt{21} - 8} = 1$$

Playing around with the expression, I found a proof which I will post as an answer.
I'm asking this question because I would like to see if there are alternative solutions which are perhaps faster / more direct / elementary / elegant / methodical / insightful etc.
 [/INST] As a bonus, we can also obtain that 
$$T=\sqrt[3]{3\sqrt{21} + 8} + \sqrt[3]{3\sqrt{21} - 8}=\sqrt{21}. $$
Indeed, we can write $$T^3 = 6\sqrt{21}+6\sqrt{21}.$$ In order to get rid of $\sqrt{21}$ we can write $T = \alpha \sqrt{21}$ to get the equation
$$21\alpha^3 - 15\alpha-6.$$
We have an obvious root $\alpha=1$, and then prove that 
$$\frac{21\alpha^3 - 15\alpha-6}{\alpha-1} = 21\alpha^2 +21\alpha+6$$does not have real roots.
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