[tex]x- \frac{1}{x}= \sqrt{5} \\ \\ (x- \frac{1}{x})^3=( \sqrt{5})^3 \\ \\ x^3-3 \cdot x^2 \cdot \frac{1}{x}+3 \cdot x \cdot \frac{1}{x^2} + \frac{1}{x^3}=5 \sqrt{5} \\ \\ x^3+ \frac{1}{x^3}-3x+ \frac{3}{x}=5 \sqrt{5} \\ \\ x^3+ \frac{1}{x^3}= 5 \sqrt{5}+3x- \frac{3}{x} \\ \\ x^3+ \frac{1}{x^3}=5 \sqrt{5}+3(x- \frac{1}{x}) \\ \\ x^3+ \frac{1}{x^3}= 5 \sqrt{5}+3 \sqrt{5} \\ \\ \boxed{x^3+ \frac{1}{x^3}=8 \sqrt{5} } [/tex]
