Question : If $x + \frac{1}{x} = \sqrt{13}$ , then what is the value of $x^{5} - \frac{1}{x^{5}}$ ?
Option 1: $169$
Option 2: $169 \sqrt{3}$
Option 3: $393$
Option 4: $507$

Question

Question : If $x = \sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}}$ , then what is the value of $(x^{2} + x - 9)$ ?

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Answer 1

Correct Answer: $393$

Solution : Given:
$x + \frac{1}{x} = \sqrt{13}$
Squaring both sides,
$(x + \frac{1}{x})^{2} = 13$
$x^{2} + \frac{1}{x^{2}} = 13 - 2$
$x^{2} + \frac{1}{x^{2}} = 11$
Also,
$x^{2} + \frac{1}{x^{2}} - 2 = 11 - 2$
$(x - \frac{1}{x})^{2} = 3^{2}$
$x - \frac{1}{x} = 3$
Cubing both sides
$x^{3} - \frac{1}{x^{3}} = 3^{3} + 3 \times 3 = 36$
Now,
$(x^{3} - \frac{1}{x^{3}}) (x^{2} + \frac{1}{x^{2}}) = 36 \times 11$
$x^{5} - \frac{1}{x^{5}} + x - \frac{1}{x} = 396$
$x^{5} - \frac{1}{x^{5}} + 3 = 396$
$\Rightarrow x^{5} - \frac{1}{x^{5}} = 393$
Hence, the correct answer is 393.

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Question : If x+1x=13, then what is the value of x5−1x5?Option 1: 169Option 2: 1693Option 3: 393Option 4: 507

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Question : If $x + \frac{1}{x} = \sqrt{13}$ , then what is the value of $x^{5} - \frac{1}{x^{5}}$ ?
Option 1: $169$
Option 2: $169 \sqrt{3}$
Option 3: $393$
Option 4: $507$