\( \Large a = \left(\sqrt{2} - 1 \right)^{\frac{1}{3}} \) \( \Large a^{3} = \sqrt{2} - 1 \) \( \Large \left(a - a^{-1}\right)^{3} + 3 \left(a - a^{-1}\right) \) = \( \Large \left(a - \frac{1}{a}\right)^{3} + 3 \left(a - \frac{1}{a}\right) \) = \( \Large a^{3} - \frac{1}{a^{3}} - 3 \left(a - \frac{1}{a}\right) + 3 \left(a - \frac{1}{a}\right) \) = \( \Large \sqrt{2} - 1 - \frac{1}{\sqrt{2}-1}\) = \( \Large \left(\sqrt{2} - 1\right) - \frac{1}{ \left(\sqrt{2-1}\right)} \times \frac{ \left(\sqrt{2} + 1\right) }{ \left(\sqrt{2} + 1\right) } \) = \( \Large \sqrt{2} - 1 - \left(\sqrt{2}+1\right) = -2 \)
