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Given that $x$ is real and $x^3+\frac{1}{x^3}=52$, find $x+\frac{1}{x}$.

 

tysm!!!

 
 Feb 23, 2021
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$\left(x+\frac{1}{x}\right)^3=x^3+\frac{1}{x^3}+3x+\frac{3}{x}=52+3\left(x+\frac{1}{x}\right)$

$x+\frac{1}{x}=k\to k^3=52+3k$. We can guess and check or use rational root theorem to find $k=4\implies\boxed{x+\frac{1}{x}=4}$. 

 
 Feb 23, 2021

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