if x+1/x= 2√5 thenprove x(x^6-1)/x^8-1 =19√5/180?

This solution is only for Positive real numbers.

x3+(1/x3)=18............(A)

or,(x3+(1/x3))2=182

or,((x3−(1/x3))2+4∗x3∗(1/x3)=324

or,((x3−(1/x3))2+4=324

or,((x3−(1/x3))2=320

or,x3−(1/x3)=√(320)

or,x3−(1/x3)=8√5............(B)

Now add A and B

A+B,

x3+(1/x3)+x3−(1/x3)=18+8√5

or,2x3=18+8√5

or,8x3=4(18+8√5)

or,(2x)3=72+32√5

or,(2x)3=5√5+3∗(√5)2∗3+3∗√5∗32+27

or,(2x)3=(3+√5)3

or,2x=3+√5

or,x=(3+√5)/2

so,x=(3+√5)/2