<s>[INST] What is the easiest way to get: $2+ \sqrt{-121} = (2+ \sqrt{-1})^3$ I was reading the book Seventeen equations have changed the world.
At some point, while the book was talking about complex numbers, I see this equation:
$2+ \sqrt{-121} = (2+ \sqrt{-1})^3$
Even if it's easy to proof the truth of this equivalence (it is enough to develop the two members),
I can't find an easy/good/fast way to obtain straight the identity. 
Can you help me? Does there exist a mathematical property that I'm missing?
 [/INST] Not sure how you approached it before.
If you want to utlise the binomial expansion
$$
(x+y)^n = \sum_{k=0}^n\left(\matrix{n\\k}\right)x^ky^{n-k}
$$
then we have
$$
\begin{align}
(x+y)^3 &= \left(\matrix{3\\0}\right)2^3 +\left(\matrix{3\\1}\right)2^2\sqrt{-1} + \left(\matrix{3\\2}\right)2^1(\sqrt{-1})^2  + \left(\matrix{3\\3}\right)(\sqrt{-1})^3 \\
&= 2^3 + 3\cdot 2^2\sqrt{-1} + 3\cdot 2\cdot-1 + \sqrt{-1}\cdot(\sqrt{-1})^2\\
&=8+12\sqrt{-1} -6 -\sqrt{-1} = 2 +11\sqrt{-1} = 2 + \sqrt{11^2\cdot - 1} \\
&=2+\sqrt{-121}
\end{align} 
$$
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