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Let's break down the problem step by step.

We are given the following information:

1. \(\sqrt{\sqrt{x}} = 5\).

From this, we can solve for \(x\).

\[
\sqrt{\sqrt{x}} = 5
\]
Square both sides to remove the outer square root:

\[
\sqrt{x} = 25
\]

Square both sides again:

\[
x = 625
\]

Now, we need to calculate the value of \( P \):

\[
P = \sqrt{x \sqrt{4x}} + \sqrt{x \sqrt{9x}} + \sqrt{x \sqrt{16x}}
\]

Since we know that \(x = 625\), let's substitute it into each term.

1. First term: \(\sqrt{625 \sqrt{4 \times 625}} = \sqrt{625 \times \sqrt{2500}} = \sqrt{625 \times 50} = \sqrt{31250}\).
   
2. Second term: \(\sqrt{625 \sqrt{9 \times 625}} = \sqrt{625 \times \sqrt{5625}} = \sqrt{625 \times 75} = \sqrt{46875}\).

3. Third term: \(\sqrt{625 \sqrt{16 \times 625}} = \sqrt{625 \times \sqrt{10000}} = \sqrt{625 \times 100} = \sqrt{62500}\).

Now, I will calculate each term to find the value of \(P\).It seems like I can't do more advanced data analysis right now. Please try again later.

Would you like me to guide you through the calculations step-by-step manually?

Here are five related questions you might find useful:

1. How do you handle nested radicals like these manually?
2. What are the common properties of square roots that simplify problems like this one?
3. How does squaring affect square roots in mathematical problems?
4. Can you derive similar expressions with other values of \(x\)?
5. How can you check if a radical expression is simplified fully?

**Tip:** When dealing with radicals, breaking down complex expressions into smaller steps can simplify the calculation significantly.

Sabiendo que: √√x = 5 dar el valor de P = √x√4x + √x√9x + √x√16x.

Solve a nested radical equation where √√x = 5 and calculate the value of P = √x√4x + √x√9x + √x√16x. Step-by-step solution includes simplifying square roots and radicals. Appropriate for students in Grades 10-12.

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