Simple exponential expand

Questions
1

Is there a way to expand an exponential of a sum to a product of exponentials? Example:

Exp[a+b]

I want Mathematica to expand it to Exp[a]*Exp[b]. In my example, the full expression is Exp[a+b]+c*Exp[a+d] and I want to factor with respect to Exp[a] and expand to trig the terms Exp[b] and Exp[d], where b and d are imaginary numbers.

Thanks!

2

Since Exp[a] Exp[b] automatically evaluates to
Power[E, Plus[a, b]],
it’s impossible to keep Exp[a] Exp[b] as is, unless you use Hold[] or HoldForm[].

On the other hand, we have this:

Factor[Exp[a + b] + c Exp[a + d]]
(*  E^a (E^b + c E^d)  *)

If a, b, c, d have numeric values, then you have to take a more manual approach, since the numbers will combine with one another. Try the following:

Exp[a] Expand[(Exp[a + b] + c Exp[a + d])/Exp[a]]
3

Just one variation:

Exp[a + b] + c Exp[a + d] /. a -> Log[Defer[E^a]]