Find the value
If \[ x = \sqrt[3]{2 + \sqrt{3}} \]
(a) 2 \quad (b) 4 \quad (c) 8 \quad (d) 9
Solution:
\[ x^3 = 2 + \sqrt{3} \]
\[ \frac{1}{x^3} = \frac{1}{2 + \sqrt{3}} \times \frac{2 – \sqrt{3}}{2 – \sqrt{3}} = 2 – \sqrt{3} \]
\[ x^3 + \frac{1}{x^3} = (2 + \sqrt{3}) + (2 – \sqrt{3}) \]
\[ = 4 \]
\[ \therefore \text{Correct Answer: } 4 \ (\text{Option b}) \]