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[tex] \left[ a^m , (a^{\dagger})^n \right][/tex]

I have managed to derive

[tex] \left[ a^m , (a^{\dagger})^n \right]= m a^{m-1} \left[ a , a^{\dagger} \right] [/tex]

(altought I should really have remebered that) but I don't know how to use that to calculate

the big thing other than by recursive application of it, and that's very messy.

Any suggestions?