Question : If $x=\sqrt[3]{2+\sqrt{3}}$, the value of $x^3+\frac{1}{x^3}$ is:
Option 1: 8
Option 2: 9
Option 3: 2
Option 4: 4
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Answer (1)
20th Jan, 2024
Correct Answer: 4
Solution :
Given:
$x=\sqrt[3]{2+\sqrt{3}}$
⇒ $x^3=2+\sqrt{3}$
So, $\frac{1}{x^3}=\frac{1}{2+\sqrt{3}}×\frac{2-\sqrt{3}}{2-\sqrt{3}}=2-\sqrt{3}$
$\therefore x^3+\frac{1}{x^3}=2+\sqrt{3}+2-\sqrt{3}=4$
Hence, the correct answer is 4.
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