Frenet formulas of $\alpha(t)=(t,t^{2},t^{3})$

Question

Given the following curve $$\alpha(t)=(t,t^{2},t^{3})$$ I gotta find its Frenet vectors. I know, for example $$N(t)=\frac{\alpha'(t)\times(\alpha''(t)\times\alpha'(t))}{||\alpha'(t)||\cdot||\alpha''(t)\times\alpha'(t)||}$$ So, in my case, I found that $$N(t)=\frac{(-4t-18t^3,12-18t^4,6t+12t^3)}{2\sqrt{(1+4t^{2}+9t^{4})\cdot(9t^{4}+9t^{2}+1)}}$$ which is a not so elegant expression. Then, what am I doing is correct? Or is there a easier way to evaluate it?