If x=2^ 1 3 +2^ 2 3 , show that x^3-6 x=6 - Vidyadip

Question

If $\text{x}=2^{\frac{1}{3}}+2^{\frac{2}{3}},$ show that $\text{x}^3-6\text{x}=6$

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Answer

$\text{x}=2^{\frac{1}{3}}+2^{\frac{2}{3}},$ $\Rightarrow\text{x}^3=\big(2^{\frac{1}{3}}+2^{\frac{2}{3}}\big)^3$ $\Rightarrow\text{x}^3=\Big(2^{\frac{1}{3}}\Big)^3+\Big(2^{\frac{2}{3}}\Big)^3+3\times2^{\frac{1}{3}}\times2^\frac{2}{3}\Big(2^\frac{1}{3}+2^\frac{2}{3}\Big)$ $\Rightarrow\text{x}^3=2+2^2+3\times2\Big(2^\frac{1}{3}+2^\frac{2}{3}\Big)$ $\Rightarrow\text{x}^3=6+6\Big(2^{\frac{1}{3}}+2^\frac{2}{3}\Big)$ $\Rightarrow\text{x}^3=6+6\text{x}$ $\text{x}^3-6\text{x}=6$

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