What is \left(x^3 + \frac{1}{x^3}\right) equal to ?
For the next two (02) items that follow : 16\left(x^4 + \frac{1}{x^4}\right) - 257 = 0
- A. \frac{65}{8} ✓
- B. \frac{63}{8}
- C. \frac{61}{8}
- D. \frac{59}{8}
Correct Answer: A. \frac{65}{8}
Explanation
From x^2 + \frac{1}{x^2} = \frac{17}{4}, add 2 to get (x + \frac{1}{x})^2 = \frac{25}{4} \implies x + \frac{1}{x} = \frac{5}{2}. Using the identity a^3 + b^3 = (a+b)^3 - 3ab(a+b), we get x^3 + \frac{1}{x^3} = (\frac{5}{2})^3 - 3(\frac{5}{2}) = \frac{125}{8} - \frac{15}{2} = \frac{125 - 60}{8} = \frac{65}{8}.
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