If e1=((2^1/3)-1)^1/3 e2=(9^-1/3)-((2/9)^1/3)+((4/9)^1/3) then prove that e1=e2.

Hello student,
given [e₁=(2^{1/3}-1)^{1/3}]
so cubing on both sides we get e₁³= [(2^{1/3}-1)] ..........(1)
Similarly [e₂=(9^{-1/3})(1-2^{1/3}+2^{2/3})]
since [(1-2^{1/3}+2^{2/3})] is in the form of a²-ab+b²=(a³+b³)/(a+b)
where a=2^1/3and b=1
so [e₂=3^{1/3}/(2^{1/3}+1)]
also cubing above equation on both sides we get
e₂³= [3/(3+3*2^{1/3}(2^{1/3}+1))]
= [1/(1+2^{1/3}+2^{2/3})]
as denominator is in the form ofa²+ab+b²=(a³-b³)/(a-b)
e23= [(2^{1/3}-1)/2-1] …...............(2)
so from 1 and 2 we can see that e1=e2