need arithmetic problem proof if  $a^2+b^2=a^3+b^3=a^4+b^4$
find a and b

This is not a proof but a visualizing aid that cannot fit in a simple "remark". The graphics below displays: 


*

*the cubic curve (c) with equation $a^2+b^2=a^3+b^3$ (in red), with an isolated point: $(0,0)$. Note the invariance by symmetry $(x,y) \rightarrow (y,x)$. 

*the quartic curve (q) with equation $a^4+b^4=a^3+b^3$ (in black), with the same isolated point. Note the invariance by different symmetries.
One "sees" the three common points $(1,0),(1,1),(0,1)$, to which must be added the isolated point $(0,0)$. 
Note: the solution by @Mingfeng Zhao amounts to intersect these curves with a straight line $b=ta$ pivoting through the origin.

