Let's go through this step-by-step.

Step 1: Understand the Given Expressions

The problem involves two parts:

1. The value of $p$: $p = 11 + \sqrt{\frac{120}{2\pi}}$
2. An expression to evaluate: $p\sqrt{p} + \frac{1}{p\sqrt{p}}$

Step 2: Simplify the Expression for $p$

First, we simplify $p$ as follows:

$p = 11 + \sqrt{\frac{120}{2\pi}} = 11 + \sqrt{\frac{60}{\pi}}$

To proceed further, let's calculate the value of $\sqrt{\frac{60}{\pi}}$.

Step 3: Calculate $\sqrt{\frac{60}{\pi}}$

Assuming $\pi \approx 3.14159$: $\sqrt{\frac{60}{\pi}} = \sqrt{\frac{60}{3.14159}} \approx \sqrt{19.0986} \approx 4.37$

So, we get: $p \approx 11 + 4.37 = 15.37$

Step 4: Substitute $p$ into the Expression

Now we need to evaluate: $p\sqrt{p} + \frac{1}{p\sqrt{p}}$

1. Calculate $p\sqrt{p}$: $\sqrt{p} = \sqrt{15.37} \approx 3.92$ $p\sqrt{p} \approx 15.37 \times 3.92 \approx 60.25$

2. Calculate $\frac{1}{p\sqrt{p}}$: $\frac{1}{p\sqrt{p}} \approx \frac{1}{60.25} \approx 0.0166$

3. Add the two results: $p\sqrt{p} + \frac{1}{p\sqrt{p}} \approx 60.25 + 0.0166 \approx 60.27$

Final Answer:

$p\sqrt{p} + \frac{1}{p\sqrt{p}} \approx 60.27$

Would you like a more detailed explanation of any step, or do you have any further questions?

Here are five additional related questions you might find interesting:

1. What is the significance of the square root operation in simplifying expressions?
2. How can this approach be extended to other similar problems involving roots and powers?
3. How would the expression change if the square root of $p$ were a different root, such as a cube root?
4. What are some real-world applications of calculating expressions like $p\sqrt{p} + \frac{1}{p\sqrt{p}}$?
5. What methods are available for approximating $\sqrt{p}$ without a calculator?

Tip: When working with square roots in fractions, simplifying the fraction as much as possible before taking the square root can make calculations easier.