Question : If $g=(2-\sqrt{3})$, what will be the value of $\left(g^3-\frac{1}{g^3}\right)$?
Option 1: $-30 \sqrt{3}$
Option 2: 52
Option 3: $30 \sqrt{3}$
Option 4: –52
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Correct Answer: $-30 \sqrt{3}$
Solution :
Given, $g=(2-\sqrt{3})$
⇒, $\frac{1}{g}=\frac{1}{2-\sqrt{3}}$
⇒ $\frac{1}{g}=\frac{2+\sqrt{3}}{(2-\sqrt{3})(2+\sqrt{3})}$
⇒ $\frac{1}{g}=2+\sqrt{3}$
$\left(g^3-\frac{1}{g^3}\right)=(g-\frac{1}{g})(g^2+\frac{1}{g^2}+g×\frac{1}{g})$
⇒ $\left(g^3-\frac{1}{g^3}\right)=(2-\sqrt{3}-(2+\sqrt{3}))((4+3-4\sqrt{3})+(4+3+4\sqrt{3})+1)$
⇒ $\left(g^3-\frac{1}{g^3}\right)=-2\sqrt{3}(15)$
⇒ $\left(g^3-\frac{1}{g^3}\right)=-30\sqrt{3}$
Hence, the correct answer is $-30\sqrt{3}$.
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