Simulate a pulse of specific power
Well, for a signal $x[k]$, the energy is:
\begin{equation}
E=dt*\sum_{k=-\infty}^{\infty}x^2[k]
\end{equation}

You want the energy to be one by dividing $x[k]$ through a value $a$, hence:
\begin{equation}
1=dt*\sum_{k=-\infty}^{\infty}(\frac{x}{a})^2[k]
\end{equation}
which by multiplying with $a^2$ becomes
\begin{equation}
a^2=dt*\sum_{k=-\infty}^{\infty}x^2[k]
\end{equation}
Comparing this to the first equation, it can be easily seen that $a^2=E$ and since $E$ is always positive, we can conclude $a=\sqrt{E}$. And that is why you divide by the square root of $E$.

From there it should be easy to see why multiplying this with $\sqrt{E'}$ makes a signal of energy $E'$.