Quote by NewtonianAlch
1. The problem statement, all variables and given/known data
Determine the Laplace transform:
g(t) = 2*e[itex]^{4t}[/itex]u(t1)
3. The attempt at a solution
Essentially we're told for a time shift we multiply the Laplace transform pair of the function (without the delay) by e[itex]^{as}[/itex]
So here a = 1 (for the delay)
The Laplace transform for e[itex]^{4t}[/itex] is [itex]\frac{1}{s + 4}[/itex]
Multiplying we should get e[itex]^{s}[/itex]([itex]\frac{2}{s + 4}[/itex])
However the answer is e[itex]^{s}[/itex]*([itex]\frac{2}{s + 4}[/itex]) * [itex]\frac{1}{e^{4}}[/itex]

It's because you must be in the form
[tex]e^{4(ta)}u(ta)[/tex]
Take a look at your exponential. It isn't timeshifted by a to the right. What you can do is use the fact that exponentials multiplied add their exponents and the fact that e^b/e^b = 1, so you can multiply by it without changing your values. So you choose e^b so that you can add its exponents and arrive to the needed shift. The e^b in the denominator is factored outside of the linear inverse Laplace operator and you go from there.