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Can I write $a^* + b^* = (a + b)^*$ ????
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NO,

we can write (a+b)*=(a*+b*)*

                               =(a*+b)*

                               =(a+b*)*

                               =(a*b*)*

                               =a*(ba*)* 

                               =b*(ab*) *

 

or

make minimal dfa for both the RE and check LHS != RHS ,infact  language both are different 

 

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$1^{st}$ one is RHS.
$2^{nd}$ one is LHS.
LHS $\neq$ RHS

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No ,because both are not same as u can see by the 1st REstring produce like ab,aabb.a,b,aa,bbetc but u can't get string like aba,aaabbabab,ababa etc .

whereas by the 2nt RE u will get all string produce by a and b.