If I simplify (3b^3) ^4 and in the end get 81b^12 how do I get there?

Everything inside of the parenthesis needs to be to the 4th root. So you can see it as:

3^4 * (b^3) ^4

3^4 = 3*3*3*3 = 81

And using rules of exponents, you need to multiply those exponents.

b ^ (3*4) = b^12

Put them together ... 81 b^12