A Non-Trivial Algebraic Number using transcendental numbers

Question

I know that there exists many non-trivial expressions which are actually rational numbers when they don't seem to be in the first place. Such as $(5+2\sqrt{7})^{\frac{1}{3}}+(5-2\sqrt{7})^{\frac{1}{3}}=1$

I wonder if there exists such non-trivial expressions involving transcendental numbers such as $e, \pi$ etc which reduced to an algebraic number. I tried to create simmilar expressions of the form $(e+a\sqrt{b})^{\frac{1}{3}}+(e-a\sqrt{b})^{\frac{1}{3}}$ but could not get anywhere.

Note-Do not suggest answers such as $e-e, \pi-\pi$!

Answer 1

Let $x$ be any nonzero real number at all. If $x$ is transcendental then surely $x+(1/x)$ and $x^3+(1/x^3)$ are, too Yet

$\dfrac{(x+(1/x))^3-(x^3+(1/x^3))}{x+(1/x)}=3$

is always true.

A similar identity can be found with $x-(1/x)$, provided $x\not\in\{0,\pm1\}$:

$\dfrac{(x-(1/x))^3-(x^3-(1/x^3))}{x-(1/x)}=-3$

This relation, and similar ones using higher odd powers, explain the apparently strange relationships between inverse hyperbolic sines of some whole numbers alluded to in Gottfried's comment.