TITLE: How do you "simplify" the sigma sign when it is raised to a power?
QUESTION [9 upvotes]: How do you simplify the following expression:
$$\left(\sum^{n}_{k=1}k \right)^2$$
I am supposed to show that 
$$\left(\sum^{n}_{k=1}k \right)^2 = \sum^{n}_{k=1}k^{3} $$
The problem is I do not really know how to manipulate the sigma sign. I know that I (probably) need to use induction somehow, but the main question is how do you "simplify" the sigma sign when it is raised to a power. Due to the problem itself I know that (most likely); $$\left(\sum^{n}_{k=1}k \right)^2 = \sum^{n}_{k=1}k^{3} $$ so is it possible to simply manipulate the LHS so that it looks like the RHS?

REPLY [6 votes]: This identity is a coincidence — it is not proven by doing general series manipulations, but instead by simply computing the left and right hand sides and confirming they're equal.