Question : If $x+\frac{1}{x}=\sqrt{3}$, then the value of $x^{3}+\frac{1}{x^{3}}$ is equal to:
Option 1: $1$
Option 2: $3\sqrt{3}$
Option 3: $0$
Option 4: $3$
Correct Answer: $0$
Solution :
Given: $x+\frac{1}{x}=\sqrt{3}$
Cubing both sides we get
$(x+\frac{1}{x})^3=(\sqrt{3})^3$
⇒ $x^3+\frac{1}{x^3}+3×x×\frac{1}{x}(x+\frac{1}{x})=(3\sqrt{3})$
⇒ $x^3+\frac{1}{x^3}=3\sqrt{3}–3(x+\frac{1}{x})$
$\because x+\frac{1}{x}=\sqrt{3}$
Thus, $x^3+\frac{1}{x^3}=3\sqrt{3}–3\sqrt{3} = 0$
Hence, the correct answer is $0$.
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