How do you "simplify" the sigma sign when it is raised to a power? 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? 

$$\begin{align}
\sum^{n}_{k=1}k&=\frac{n(n+1)}{2}\\
\left(\sum^{n}_{k=1}k\right)^2&=\left(\frac{n(n+1)}{2}\right)^2\\
\left(\frac{n(n+1)}{2}\right)^2&=\frac{n^2(n+1)^2}{4}\\
\sum^{n}_{k=1}k^3&=\frac{n^2(n+1)^2}{4}\\
\therefore \left(\sum^{n}_{k=1}k\right)^2&=\sum^{n}_{k=1}k^3
\end{align} $$
