Given : \( \Large x+\frac{1}{3}=\sqrt{3} \)

Taking cube on both sides, we have

\( \Large \left(x+\frac{1}{x}\right)^{3}= \left(\sqrt{3}\right)^{3} \)

=> \( \Large x^{3} + \frac{1}{x^{3}} + 3x^{2}\frac{1}{x} + 3x\frac{1}{x^{2}} = 3\sqrt{3} \)

=> \( \Large x^{3}+\frac{1}{x^{3}}+3 \left(x+\frac{1}{x}\right)=3\sqrt{3} \)

=> \( \Large x^{3}+\frac{1}{x^{3}}+3 \times \sqrt{3} = 3\sqrt{3} \)

=> \( \Large x^{3}+\frac{1}{x^{3}} = 0 \)