Ramanujan's Surd : Proof required. I came across a problem on the web, it is as follows:
Question:If $  x  $, $  y  $ and $  z  $ are rational numbers such that
$  \sqrt[3]{\sqrt[3]{2}-1} = \sqrt[3]{x}+\sqrt[3]{y}+\sqrt[3]{z}$ then find $ x,y,z $.
From Wikipedia|Nested Radicals|Some identities by Ramanujan, it is clear that the values of $x,y,\text{and} \ z$ is $\sqrt[3]{\frac{1}{9}}$,$-\sqrt[3]{\frac{2}{9}}$ and $\sqrt[3]{\frac{4}{9}}$ respectively. 
However, even after going through the references from the wikipedia and from the web [2], [3], [4]; I still fail at understanding it. 
Can someone please show me the proof of this result in an intuitive way?
My approach in attacking this problem  is as follows, 
Let, $y = \sqrt[3]{\sqrt[3]{2}-1}$
Cubing both sides,
$$
y^{3} = \sqrt[3]{2}-1 \\
\implies y^{3}+1=\sqrt[3]{2} \\
\implies (y^{3}+1)^{3}= 2 
$$
Now, using the identity : $(a+b)^3=a^3 + 3a^{2}b + 3ab^{2} + b^{3}$
$$
y^{9} + 3y^{6}+3y^{3}+1 = 2 \\
\implies y^{9} + 3y^{6}+3y^{3} - 1 = 0
$$
Now, I get the cubic equation using the substitution, $y^{3} = m$
$$
m^{3} + 3m^{2}+3m - 1 = 0
$$
Seems like I get lost in my own web.
Can someone help me out?

You want a proof of the identity $$\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{\frac{1}{9}}-\sqrt[3]{\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}$$ which can be written as $$\sqrt[3]{x-1}=\frac{1-x+x^{2}}{\sqrt[3]{9}}$$ where $x=\sqrt[3]{2}$. Thus we let the RHS be denoted by $y$ and show that $y^{3}=x-1$. Since $x, y$ are real this proves the desired identity. 
Now we can see that $$y^{3}=\frac{(1-x+x^{2})^{3}}{9}=\frac{(x^{3}+1)^{3}}{9(x+1)^{3}}=\frac{3}{x^{3}+3x^{2}+3x+1}=\frac{1}{1+x+x^{2}}=\frac{x-1}{x^{3}-1}=x-1$$ In the above algebraic manipulation $x^{3}$ has been replaced with $2$.
Ramanujan was expert in algebraic manipulation of numbers as well as symbolic expression so much so as to make MAPLE and MACSYMA almost unnecessary and he obtained most of such denesting of radicals as special cases of general algebraic identities. It was not exactly like solving equations to find value of $x, y,z$ as you ask in the question. 
