Question

Find the inverse transform of each of the followings f(s) = 7/(s+3)^3

Answer 1

normally to find the inverse switch the x and y then solve for y
here f(s) = y and s = x

Answer 2

Recall a few things:
~Exponential causes a shift
~Transform of t^n is going to be helpful here as well\[\large\rm \mathscr L\{t^n\}=\frac{n!}{s^{n+1}}\]

Answer 3

If we throw an exponential into the mix,\[\large\rm \mathscr L\{e^{at}t^n\}=\frac{n!}{(s-a)^{n+1}}\]it just shifts s by the amount a,

Answer 4

\[\large\rm \mathscr L\{e^{\color{orangered}{a}t}\color{royalblue}{t^n}\}=\color{royalblue}{\frac{n!}{(s-\color{orangered}{a})^{n+1}}}\]
I'm going to rewrite your expression in a clever way,
hopefully you can see what's going on :)\[\large\rm \frac{7}{(s+3)^3}\quad=\quad\frac{7}{2}\cdot\frac{2}{(s-(-3))^{2+1}}\quad=\quad \frac{7}{2}\cdot\color{royalblue}{\frac{2!}{(s-\color{orangered}{(-3)})^{2+1}}}\]

Answer 5

Hopefully that didn't make it too confusing.

Answer 6

how to get this to what you have or is this incorrect?