Question

If x + (1/x) = √13, then what is the value of x5 – (1/x5)? in detail

Answer 1

I have attached the answer.

plz let me know if any further clarification is required.

Answer 2

Thanks for asking the question. Here is your answer:

Given in the question,

$x+\frac{1}{x} = \sqrt{13}$  and $x^{5} + \frac{1}{x^{5}}=?$

On squaring both sides we get,

$(x+\frac{1}{x})^{2} = (\sqrt{13})^{2}$

or, $x^{2} + 2.x.\frac{1}{x} + (\frac{1}{x})^{2}  = 13$

or, $x^{2} + 2+\frac{1}{x^{2} } = 13$

or, $x^{2} +\frac{1}{x^{2}}= 13-2=11$ ------------------(1)

On cubing both sides of the equation we get,

$(x+\frac{1}{x})^{3} = (\sqrt{13} )^{3}$

or, $x^{3} + \frac{1}{x^{3} }+ 3.x.\frac{1}{x}(x+\frac{1}{x} )= 13\sqrt{13}$

or, $x^{3}+\frac{1}{x^{3} } + 3\sqrt{13}= 13\sqrt{13}$ ( given in the question, ∵$x+\frac{1}{x} = \sqrt{13}$ )

or, $x^{3} +\frac{1}{x^{3} }=13\sqrt{13}-3\sqrt{13}$

or,   $x^{3} + \frac{1}{x^{3} }= 10\sqrt{13}  ------------------(2)$

Multiplication from (1) to (2) we get,

$x^{2} + \frac{1}{x^{2} } (x^{3} +\frac{1}{x^{3} } )= 11 (10\sqrt{13})$

or, $x^{2} (x^{3}+\frac{1}{x}^{3})+\frac{1}{x^{2} }(x^{3}+\frac{1}{x}^{3})$= $110\sqrt{13}$

or, $x^{5}+\frac{1}{x}+\frac{x^{3} }{x^{2} } + \frac{1}{x^{5} } = 110\sqrt{13}$

or, $x^{5} + \frac{1}{x^{5} }= 110\sqrt{13} - (\frac{1}{x} + x)$

or, $x^{5} + \frac{1}{x^{5} }= 109\sqrt{13}$ ( given in the question, ∵$x+\frac{1}{x} = \sqrt{13}$ )

Hence, the solution is  $x^{5} + \frac{1}{x^{5} }= 109\sqrt{13}$