Draw a DFA that accepts ((aa*)*b)*
I recommend you first try to understand the language denoted by the regular expression. Once done this can make it easier to build the DFA directly . If you can not think of how to build the DFA you can always turn your regular expression into an NFA and then the NFA into a DFA(Convert regular expression to DFA).

The strings denoted by your regular expression are:


   Empty string or a concatenation of strings formed by a chain of zero or more $a's$ followed by a $b$. This mean that your initial state should be a final state too and a word in this language "always"(except  $\epsilon$) finish with a letter $b$ so your DFA can't transition to a final state consuming a letter $a$. I hope those hints can help you to generate your DFA directly.