Question
If $\text{x}=2^{\frac{1}{3}}+2^{\frac{2}{3}},$ show that $\text{x}^3-6\text{x}=6$
✓
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$
$\Rightarrow\text{x}^3=\Big(2^{\frac{1}{3}}\Big)^3+\Big(2^{\frac{2}{3}}\Big)^3+3\times2^{\frac{1}{3}}\times2^\frac{2}{3}\Big(2^\frac{1}{3}+2^\frac{2}{3}\Big)$
$\Rightarrow\text{x}^3=2+2^2+3\times2\Big(2^\frac{1}{3}+2^\frac{2}{3}\Big)$
$\Rightarrow\text{x}^3=6+6\Big(2^{\frac{1}{3}}+2^\frac{2}{3}\Big)$
$\Rightarrow\text{x}^3=6+6\text{x}$
$\text{x}^3-6\text{x}=6$
$\Rightarrow\text{x}^3=\Big(2^{\frac{1}{3}}\Big)^3+\Big(2^{\frac{2}{3}}\Big)^3+3\times2^{\frac{1}{3}}\times2^\frac{2}{3}\Big(2^\frac{1}{3}+2^\frac{2}{3}\Big)$
$\Rightarrow\text{x}^3=2+2^2+3\times2\Big(2^\frac{1}{3}+2^\frac{2}{3}\Big)$
$\Rightarrow\text{x}^3=6+6\Big(2^{\frac{1}{3}}+2^\frac{2}{3}\Big)$
$\Rightarrow\text{x}^3=6+6\text{x}$
$\text{x}^3-6\text{x}=6$
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