Question

x^3000+(1/x)^(3000)=?
WHEN x=2+3^(1/2)

Answer 1

$we \: have \\ x = 2 + {3}^{ \frac{1}{2} } = 2 + \sqrt{3} \\ now \\ {x}^{3000} + {( \frac{1}{2 + \sqrt{3} })}^{3000} = {(2 + \sqrt{3} )}^{3000} + { (\frac{2 - \sqrt{3} }{4 - 2 \sqrt{3} + 2 \sqrt{3} - 3 } )}^{3000} \\ = {(2 + \sqrt{3} )}^{3000} + {( \frac{2 - \sqrt{3} }{1} )}^{3000} \\ = {(2 + \sqrt{3} )}^{3000} + {(2 - \sqrt{3} )}^{3000}$