Given \(x = 2^{1/3} + 2^{2/3}\), show that \(x^3 – 6x = 6\)
Solution
\[ x = 2^{1/3} + 2^{2/3} \]
\[ x^3 = (2^{1/3} + 2^{2/3})^3 \]
\[ = (2^{1/3})^3 + (2^{2/3})^3 + 3 \cdot 2^{1/3} \cdot 2^{2/3}(2^{1/3} + 2^{2/3}) \]
\[ = 2 + 4 + 3 \cdot 2 (2^{1/3} + 2^{2/3}) \]
\[ = 6 + 6x \]
\[ \Rightarrow x^3 = 6 + 6x \]
\[ \Rightarrow x^3 – 6x = 6 \]