The Energy Equation I've been studying the energy equation in relativistic motion  $E= \frac{mc^2}{\sqrt{1-v^2/c^2}}$, which can be expanded as $$E = mc^2 + \frac{1}{2} mv^2\text{ + some other terms.}$$
I'm curious about those other terms.  It kind of surprised me that they are there.  For example the third term in the expansion would be $\frac{3}{8} \frac{v^4}{c^2}$ which would become significant, I'm assuming, somewhere around $v = \sqrt{c}$.
Just wondering what the implications of this is?  Do we need to account for these?  I thought a $v^4$ term would stop velocity being rotationaly invariant?
 A: 
Do we need to account for these?

Yes: if the speed of an object is not small compared to the speed of light, the remaining terms become non-negligible and you have to use the exact expression for the energy,
$$
E=\frac{mc^2}{\sqrt{1-(v/c)^2}}
$$
Sometimes, instead of using the exact expression it is convenient to keep the Newtonian term $\frac{p^2}{2m}$ and the next term in the expansion,
$$
-\frac{p^4}{8m^3c^2}
$$
This is used, for example, in the fine structure of the hydrogen atom, where the Newtonian term is the dominant contribution but the second order term is measurable.

I thought a $v^4$ term would stop velocity being rotationally invariant?

Not really: the quartic term is $v^4=(\boldsymbol v\cdot\boldsymbol v)^2$, and as $\boldsymbol v\cdot\boldsymbol v$ is a scalar, so is its square. The exact expression of $E$ is rotationally invariant, and so is the expansion parameter $v/c$, which means that all the terms in the Taylor expansion are scalars.

For example the third term in the expansion would be $\frac{3}{8} \frac{v^4}{c^2}$ which would become significant, I'm assuming, somewhere around $v=\sqrt c$.

The relation $v=\sqrt{c}$ doen't make sense, because both $v$ and $c$ have units of velocity.
On the other hand, we could say that the third term becomes relevant when it is, for example, at least ten times smaller than the second term:
$$
\frac{3}{8} \frac{v^4}{c^2}\ge 0.1\ \frac{1}{2}v^2
$$
which is the same as $v\ge \sqrt{2/15}\ c\approx0.37c$.
