<s>[INST] If $\sqrt[3]x-\sqrt[3]{x-14}=2$, find $x - 1/x$ 
$$ \sqrt[3]x-\sqrt[3]{x-14}=2$$
  Then find $x - \frac{1}{x} = ?$ 

I've tried to take third power of both sides but I couldn't find anything. 
 [/INST] Hint: $(a-b)^3=a^3-b^3-3ab(a-b)\,$ so: $$8 = \left(\sqrt[3]{x}-\sqrt[3]{x-14}\right)^3=x - (x-14) - 3\cdot\sqrt[3]{x\,(x-14)} \cdot 2 = 14 - 6 \sqrt[3]{x\,(x-14)}$$ It follows that: $$x(x-14) = 1 \iff x^2-14x - 1 = 0 \iff x - \frac{1}{x} - 14=0$$
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