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: $169\sqrt3$
Option 3: $393$
Option 4: $507$
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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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