<s>[INST] Finding $x-\frac{1}{x}$ if $x^{3}-\frac{1}{x^{3}}=76$. If $x$ is a real number that satisfies $x^{3}-\frac{1}{x^{3}}=76$, determine the value of $x-\frac{1}{x}$.
How can I solve this problem?
 [/INST] $a^3-b^3=(a-b)(a^2+ab+b^2)$ and note that $(x-\frac{1}{x})^2 =x^2+\frac{1}{x^2}-2$ then if we set $(x-\frac{1}{x})=t$ we have $t(t^2+3)=76$ now find $t$.
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