Deriving relativistic energy from momentum and Lagrangian

[genericusrnme]

I've been reading through a book on relativity and I came across this

$\mathcal{E}=p . v - L$
$\mathcal{E} =\frac{m v}{\sqrt{1-\frac{v^2}{c^2}}}.v + m c^2\sqrt{1-\frac{v^2}{c^2}}$

I know this part well enough but then the book arrives at

$\mathcal{E}=\frac{m c^2}{\sqrt{1-\frac{v^2}{c^2}}}$

How did that happen?
All I can get is this

$\mathcal{E} =\frac{m v^2}{\sqrt{1-\frac{v^2}{c^2}}} + m c^2\sqrt{1-\frac{v^2}{c^2}}\neq \frac{m c^2}{\sqrt{1-\frac{v^2}{c^2}}}$

What am I doing wrong here?

Thanks in advance

 

[Parlyne]

Multiply the second term by
$$\frac{\sqrt{1-\frac{v^2}{c^2}}}{\sqrt{1-\frac{v^2}{c^2}}}$$
and I think you'll find that it works.

 

[genericusrnme]

Ah, thanks Parlyne
I feel like a tool now :L