formal language and automata theory
please answer I want the proof thank you
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Your title expression $(a+b)^\ast=a^\ast(ba^\ast)^\ast$ is not true, since the right side allows the instance $b$ alone, but every nonempty instance on the left must have at least one $a$.
Meanwhile, the expression in the body of your question $(a+b)=a(ba^\ast)^\ast$ is not true, since all instances of the left expression have only one $b$, but on the right, we can have $abbbb$.
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