Lorentz Invariance & Finding Lambda Expression

[parton]

Homework Statement

I have two four vectors v and w with $$v^{2} = m^{2} > 0, v_{0} > 0$$ and $$w^{2} > m^{2}, w_{0} > 0$$. Now we consider a system with
$$w' = (w_{0}', \vec{0})$$ and $$v' = (v_{0}', \vec{v} \, ')$$ and in addition we consider the quantity $$\lambda = \vert \vec{v}' \vert \, \sqrt{ w_{0}'^{2} - m^{2}}$$. Now I should find a Lorentz invariant expression of $$\lambda$$ only using the invariants $$v^{2}, w^{2}, vw$$.

Homework Equations

The Attempt at a Solution

I think I've found a solution: $$t = \sqrt{\dfrac{(vw)^{2} - v^{2} w^{2}}{v^{2}} (v^{2} - w^{2})}$$.
But I'm not really sure if this "solution" is really Lorentz invariant (my problem is the square root). Could anyone confirm this solution or is there any mistake?

 

[Avodyne]

The square root is OK as long as what you're taking the root of is positive. I didn't check your math, but the answer should look something like this.