A question on ordinal arithmetic.

Question

I have to order these two ordinals and I was just wondering if I have done it correctly.

$\omega^\omega + \omega^3$ and $\omega + \omega^3 +\omega^\omega$

I have worked out that $\omega + \omega^3 + \omega^\omega = \omega^\omega$ and that $\omega^\omega + \omega^3 = \omega^{\omega + 3}$

so $\omega^\omega + \omega^3 > \omega + \omega^3 +\omega^\omega$

Is this correct?