If x^2- 5x +1=0 then , find the value of

x^10 + 1 / x^5.

kindly give the method of solving.

x^5  +  1/x^5 

Product of roots  ab = 1 => a=1/b

a^5  +  1/a^5  =  a^5 +  b^5 = ((5 + root(21) )^5)/32  +  ((5 - root(21))^5)/32

= (1/32) * [(5 + root(21))^5 + (5-root(21))^5]

Apply binomial expansion all even terms of each expansion gets cancelled and odd terms get doubled

= (2/32) * [ (1)(5)^5 + (10)(5)^3(21) + 5(5)(21)^2]

= (1/16)*[3125 + 26250 + 11025]

=40500/16

=10125/2