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 \( n \).

Explore the Laplace transform of t^(4/3) using the Gamma function. This problem covers the Laplace transform formula for fractional powers and evaluates the Gamma function numerically to provide a detailed solution. Suitable for undergraduate math students.

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