Question

if x=(√2+1)to the power -1/3 prove that (x-x to the power -1) ³ +3(x-x to the power -1) + 2=0

Answer 1

x=(√2+1)to the power -1/3 prove that (x-x to the power -1) ³x=(√2+1)^( −1/3)
 To prove (x-x^ -1) ³ +3(x-x ^ -1) + 2=0
LHS=(x-1/x) ³ +3(x-1/x ) + 2=x³-3(x)(1/x)(x-1/x)-1/x³+3(x-1/x )+2
=x³-3(x-1/x)-1/x³+3(x-1/x )+2
=x³-3(x-1/x)+3(x-1/x )-1/x³+2
=x³-1/x³+2         
x³={(√2+1)^( −1/3)}³=(√2+1)^( −1) =1/(√2+1) =1(√2−1)/{(√2+1)(√2−1)}
=1(√2−1)/{2−1} =(√2−1)
and 1/x³=(√2+1) 
Hence  x³-1/x³+2 =(√2−1)−(√2+1)+2
=√2−1−√2−1+2 =−2+2=0=RHS  Hence proved