The given problem asks to find the value of the following expression:

$\sqrt{ \left( \frac{\frac{2}{\sqrt{3}} - 1}{5} \right)^2 } - \sqrt{ \left( \frac{4}{1 + \frac{1}{\sqrt{3}}} \right)^2 } = ?$

Let's break it down step by step:

First term:

$\sqrt{ \left( \frac{\frac{2}{\sqrt{3}} - 1}{5} \right)^2 }$

1. Start by simplifying $\frac{2}{\sqrt{3}} - 1$: $\frac{2}{\sqrt{3}} \approx 1.1547, \quad 1.1547 - 1 = 0.1547$
2. Now divide by 5: $\frac{0.1547}{5} = 0.03094$
3. Square this result: $(0.03094)^2 = 0.0009577$
4. Taking the square root: $\sqrt{0.0009577} \approx 0.03094$

Second term:

$\sqrt{ \left( \frac{4}{1 + \frac{1}{\sqrt{3}}} \right)^2 }$

1. Simplify $1 + \frac{1}{\sqrt{3}}$: $\frac{1}{\sqrt{3}} \approx 0.5774, \quad 1 + 0.5774 = 1.5774$
2. Now divide 4 by 1.5774: $\frac{4}{1.5774} \approx 2.536$
3. Square this result: $(2.536)^2 = 6.428$
4. Taking the square root: $\sqrt{6.428} \approx 2.536$

Final calculation:

Now subtract the two results: $0.03094 - 2.536 = -2.505$

So, the value of the expression is approximately $-2.505$.

Would you like further details or clarifications on any of the steps? Here are some related questions:

1. How is the expression $\frac{2}{\sqrt{3}}$ simplified?
2. Can you explain why squaring and taking the square root cancel each other out?
3. What are other methods to simplify complex fractions involving roots?
4. How can you verify the approximation accuracy for square roots?
5. What are some strategies for handling algebraic expressions with multiple terms like this?

Tip: Always simplify within the parentheses first before applying operations like squaring or taking the square root!