LaplaceTransform[At^-a + Bt^-b, t, s]
A s^(-1 + a) Gamma[1 - a] + B s^(-1 + b) Gamma[1 - b]

Simplify[A s^(-1 + a) Gamma[1 - a] + B s^(-1 + b) Gamma[1 - b]]
    (A s^a Gamma[1 - a] + B s^b Gamma[1 - b])/s

How do I then assign (a+b=1).

Replace[a, a -> 1 - b]
1 - b

and then substitute into equation and reduce with this identity

simplify[Gamma[1 - a] Gamma[a]]
    \[Pi] Csc[a \[Pi]]

This is from an old Schaum's outline, by hand i get the s powers reversed with its corresponding capital, as opposed to the outline

{(As^-a + Bs^-b) Sqrt[\[Pi]] Sqrt[Csc[\[Alpha]\[Pi]]]}