Given: AB=B and BA=A
lets break the question as :
(A+B)^{5} = (A+B)^{2} * (A+B)^{2} * (A+B)
=(A^{2}+B^{2}+AB+BA)(A^{2}+B^{2}+AB+BA)(A+B)[email protected]
-------------------------------------------------------------------------------------------------
(Since (A+B)^{2}= (A^{2}+AB+BA+B^{2}) = (A*A + AB+BA+B*A)
=(A*BA+AB+BA+B*AB) = ((A*B)A+AB+BA+(B*A)B)
=(BA+AB+BA+AB) = 2(A+B)[email protected]
-------------------------------------------------------------------------------------------------
Puting value of equation @2 in @1 we get :->
=2(A+B)*2(A+B)*(A+B) = 4(A+B)^{2}*(A+B) = 8(A+B)(A+B)
=16(A+B)_{ answer}