Almost correct: multiplication of b and x should be written as a * or as a space:
Manipulate[ Plot[{a*Exp[b x]}, {x, 0, 75}, PlotRange -> {{0, 100}, {0, 200}}], {a, 15, 25}, {b, 0.02, 0.03}]
or
Manipulate[ Plot[{a*Exp[b*x]}, {x, 0, 75}, PlotRange -> {{0, 100}, {0, 200}}], {a, 15, 25}, {b, 0.02, 0.03}]