Firstly , let's talk about (a+b)* , you might be knowing that it's a Universal language to generate all strings over {a,b}.
Now , to prove (a+b)*=(a*b*)*,
- we will start with the minimal string that is null string.Both of them produces it with *.
- 1-Length strings, a,b can also be produced by both.
- 2-Length strings aa,ab,ba,bb can also be produced by both.
- n-Length strings aaaaa.......,ababa......,bbbbb.......... can be produced by both.
like wise , take one string that is definitely produced by(a+b)*, just compare and conclude whether the same can be produced by (a*b*).
Conclusion: After a decent number of comparisons we can conclude that both the expressions are producing same strings.