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Algebra Level 5

$\left(z + \dfrac{1}{z}\right) + \left(z^2 + \dfrac{1}{z^2}\right)^2 + \left(z^3 + \dfrac{1}{z^3}\right)^3 + \cdots + \left(z^{10} + \dfrac{1}{z^{10}}\right)^{10}$

If $z$ is a complex number satisfying $z^2 + z + 1 = 0$, then what is the value of the expression above?

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