Problem Analysis:

We are given:

1. $P$ is a perfect cube ($P = n^3$ for some integer $n$).
2. $P > 14400$.
3. The task is to find the smallest $P$, and then compute the cube root of $P$.

Step 1: Solve for the smallest $P$

To satisfy $P > 14400$, start by finding the cube root of 14400: $\sqrt[3]{14400} \approx 24.76$ This means the smallest integer $n$ such that $n^3 > 14400$ is $n = 25$.

$P = 25^3 = 15625$

Step 2: Compute the prime factorization of 15625

To express 15625 as a product of its prime numbers: $15625 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^6$

Step 3: Find the cube root of $P$

The cube root of $P = 5^6$ is: [ \sqrt[3]{15625} = \sqrt[3]{5^6} = 5^2 = 25 ]

Final Answer:

• Smallest $P > 14400$ is $P = 15625$.
• The cube root of $P$ is $25$.

Would you like me to elaborate further or explain any specific part?

Here are 5 related questions to expand your understanding:

1. What is the general formula for finding the $n$-th root of a number expressed as prime factors?
2. Why is 14400 not a perfect cube, and how can you quickly determine this?
3. How do you compute the cube root of a large number without a calculator?
4. How are perfect cubes useful in solving real-world mathematical problems?
5. Can you describe a method to verify $n^3 > x$ without approximating roots?

Tip: When solving for perfect powers, always consider their prime factorization to simplify cube root calculations.