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To solve the Laplace transform of the given piecewise function \( f(t) \), we can break the process into steps.

The function is defined as:
\[
f(t) = 
\begin{cases} 
0, & t < 6 \\
(t - 6)^5, & t \geq 6
\end{cases}
\]

### Step 1: Use the Heaviside (Unit Step) Function

We can express the function in terms of the Heaviside step function \( u(t - 6) \). The given function can be written as:
\[
f(t) = (t - 6)^5 u(t - 6)
\]

### Step 2: Laplace Transform of Shifted Function

The Laplace transform of a shifted function \( (t - a)^n u(t - a) \) is given by the formula:
\[
\mathcal{L}\{(t - a)^n u(t - a)\} = \frac{n!}{s^{n+1}} e^{-as}
\]
where \( a = 6 \) and \( n = 5 \).

### Step 3: Apply the Formula

Substitute the values \( n = 5 \) and \( a = 6 \) into the formula:
\[
\mathcal{L}\{(t - 6)^5 u(t - 6)\} = \frac{5!}{s^{5+1}} e^{-6s}
\]

Simplify \( 5! \):
\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

Thus, the Laplace transform is:
\[
\mathcal{L}\{f(t)\} = \frac{120}{s^6} e^{-6s}
\]

### Final Answer:
\[
\mathcal{L}\{f(t)\} = \frac{120 e^{-6s}}{s^6}
\]

Would you like further details or have any specific questions about this solution?

Here are 5 related questions for further exploration:
1. How do we apply the Heaviside function in more complex Laplace transforms?
2. What is the Laplace transform of \( (t - a)^n u(t - a) \) for general \( n \) and \( a \)?
3. How can Laplace transforms handle more intricate piecewise functions?
4. Can you explain the role of shifting the function by a constant \( a \) in Laplace transforms?
5. How does the Laplace transform change when dealing with discontinuous functions?

**Tip:** When using the Heaviside function in Laplace transforms, it helps to carefully identify the shift in time to avoid mistakes in applying the exponential term \( e^{-as} \).

Find the Laplace transform of f(t) = { 0, t < 6 ; (t - 6)^5, t ≥ 6 }

Explore the Laplace transform of a piecewise function using the Heaviside step function. The function f(t) = (t - 6)^5 for t ≥ 6 is transformed step-by-step, with formulas for shifted Laplace transforms and detailed factorial calculations.

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