To find the Laplace transform of $t^{4/3}$, we use the general formula for the Laplace transform of $t^n$, where $n > -1$:

$\mathcal{L}\{t^n\} = \frac{\Gamma(n+1)}{s^{n+1}}$ Here, $\Gamma(n+1)$ is the Gamma function, which generalizes the factorial. For $t^{4/3}$, we have $n = 4/3$.

Thus, the Laplace transform is:

$\mathcal{L}\{t^{4/3}\} = \frac{\Gamma\left(\frac{4}{3} + 1\right)}{s^{\frac{4}{3} + 1}} = \frac{\Gamma\left(\frac{7}{3}\right)}{s^{7/3}}$

The Gamma function $\Gamma\left(\frac{7}{3}\right)$ can be evaluated numerically. Using properties of the Gamma function:

$\Gamma\left(\frac{7}{3}\right) \approx 1.504$

So, the Laplace transform of $t^{4/3}$ is approximately:

$\mathcal{L}\{t^{4/3}\} \approx \frac{1.504}{s^{7/3}}$

Would you like further details on how to compute the Gamma function or explore its properties?

Here are five related questions:

1. What is the Laplace transform of $t^n$ for other fractional powers?
2. How does the Gamma function generalize factorials for non-integer values?
3. What is the significance of the Gamma function in probability and statistics?
4. Can you find the inverse Laplace transform of $\frac{1}{s^{7/3}}$?
5. How does the Laplace transform help in solving differential equations?

Tip: The Gamma function $\Gamma(n)$ is an extension of the factorial function for real and complex numbers, with $\Gamma(n) = (n-1)!$ for positive integers $n$.