(m+9)^1/3=3+(m-9)^1/3

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 this problem bugged me a lot!!  The basic to solve root type equations is to get rid of the roots somehow.... I did this by cubing both sides of the equation.  this will allow you to get rid of the m term on both sides and what you'll end up with is an equation that you can transform by letting x = (m-9)^1/3.  If you do that then the equation ends up being a quadratic in x and you can find two answers via the quadratic formula.  Once you find these answers for x (-0.3820 and -2.618034) then you can solve for m and you get 2 answers; 8.944272 and -8.944272.  Plugging these into the equation checks. Hope this helps.