Question : If $c+ \frac{1}{c} =\sqrt{3}$, then the value of $c^{3}+ \frac{1}{c^{3}}$ is equal to:
Option 1: 0
Option 2: $\sqrt[3]{3}$
Option 3: $\frac{1}{\sqrt{3}}$
Option 4: $\sqrt[6]{3}$
Correct Answer: 0
Solution :
Given: $c+ \frac{1}{c} =\sqrt{3}$
$⇒(c+ \frac{1}{c})^{3} =(\sqrt{3})^{3}$
$⇒c^{3}+ \frac{1}{c^{3}}+3×c×\frac{1}{c}(c+ \frac{1}{c}) =3\sqrt{3}$
$⇒c^{3}+ \frac{1}{c^{3}}+3×\sqrt{3}=3\sqrt{3}$
$⇒c^{3}+ \frac{1}{c^{3}}=3\sqrt{3} -3\sqrt{3}$
$\therefore c^{3}+ \frac{1}{c^{3}}=0$
Hence, the correct answer is 0.
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