To find the inverse Laplace transform of $\frac{1}{(s+4)^{3/2}}$, we'll need to refer to the general rules and properties of the inverse Laplace transform. Specifically, this involves a function that resembles a fractional power of $s$.

General Approach:

One key property of the inverse Laplace transform involves recognizing shifted forms like $(s + a)$. A related standard form is:

$\mathcal{L}^{-1} \left( \frac{1}{s^{\alpha}} \right) = \frac{t^{\alpha-1}}{\Gamma(\alpha)}, \quad \text{for } \alpha > 0,$ where $\Gamma(\alpha)$ is the Gamma function.

However, the given function has a shift $s + 4$ and a fractional exponent. We can rewrite the given function as:

$\frac{1}{(s+4)^{3/2}}.$

Step 1: Use the shift theorem

The shift theorem states: $\mathcal{L}^{-1} \left( f(s + a) \right) = e^{-at} \mathcal{L}^{-1} \left( f(s) \right),$ where $a = 4$ in our case.

Thus, we can first find the inverse Laplace of $\frac{1}{s^{3/2}}$, then apply the shift theorem.

Step 2: Inverse Laplace of $\frac{1}{s^{3/2}}$

From the standard Laplace transform table: $\mathcal{L}^{-1} \left( \frac{1}{s^{\alpha}} \right) = \frac{t^{\alpha - 1}}{\Gamma(\alpha)}, \quad \alpha = \frac{3}{2}.$ So, for $\frac{1}{s^{3/2}}$: $\mathcal{L}^{-1} \left( \frac{1}{s^{3/2}} \right) = \frac{t^{\frac{1}{2}}}{\Gamma\left( \frac{3}{2} \right)}.$

Since $\Gamma\left( \frac{3}{2} \right) = \frac{1}{2} \sqrt{\pi}$, we get: $\mathcal{L}^{-1} \left( \frac{1}{s^{3/2}} \right) = \frac{2}{\sqrt{\pi}} t^{\frac{1}{2}}.$

Step 3: Apply the shift theorem

Now, apply the shift theorem with $a = 4$: $\mathcal{L}^{-1} \left( \frac{1}{(s+4)^{3/2}} \right) = e^{-4t} \cdot \frac{2}{\sqrt{\pi}} t^{\frac{1}{2}}.$

Final Answer:

The inverse Laplace transform of $\frac{1}{(s+4)^{3/2}}$ is: $\mathcal{L}^{-1} \left( \frac{1}{(s+4)^{3/2}} \right) = \frac{2}{\sqrt{\pi}} t^{\frac{1}{2}} e^{-4t}.$

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Let me know if you want more details on any step or if you have any further questions!

Here are five related questions that might interest you:

1. What is the Laplace transform of $e^{-at} t^n$?
2. How do you find the inverse Laplace of more complicated expressions?
3. What is the Gamma function and how is it used in Laplace transforms?
4. How does the shift theorem work in Laplace transforms?
5. Can inverse Laplace transforms be done for functions involving higher powers or other complex forms?

Tip: When dealing with fractional exponents in Laplace transforms, the Gamma function often plays a crucial role in the solution process.