Inversion of Laplace transformation of power functions

Questions
1

Hello there,
I have question about Laplace transformation then inverting it back to time domain, as follows:

E(t) = x(t)/y(t) = a1*t^ -b1

then the equation is transformed to Laplace domain, and invert it to back time domain, which gives the original equation (a1*t^ -b1).

However inverting each of X(t), and y(t) to Laplace domain separately, and perform the calculation as:

E(s) = x(s)/y(s) = a2*s^b2

then invert it back to time domain doesn’t give a correct answer or the original power equation of E(t)?

Thanks!