The given expression is:

$\sqrt[3]{7 - 4\sqrt{3}} \cdot \sqrt{2 + \sqrt{3}}$

Let me help simplify and solve it step by step.

Steps to solve:

1. Simplifying the cube root expression:

   • The term $7 - 4\sqrt{3}$ inside the cube root doesn't seem to simplify directly to a known perfect cube. It may involve approximation or require a special identity.

2. Simplifying the square root expression:

   • Similarly, $\sqrt{2 + \sqrt{3}}$ could be simplified if there's a known identity, but otherwise, it could be left in its current form.

To proceed further, we can evaluate these terms numerically.

Let me calculate the numerical value.The numerical value of the given expression $\sqrt[3]{7 - 4\sqrt{3}} \cdot \sqrt{2 + \sqrt{3}}$ is approximately:

$0.803$

Would you like further details or a breakdown of any part of the solution?

Here are five related questions to explore further:

1. How do we find cube roots of more complex expressions?
2. Can the square root of $2 + \sqrt{3}$ be simplified further without using a calculator?
3. What are the steps for simplifying nested radicals in general?
4. How do numerical approximations of radicals differ from exact values?
5. What are common identities involving cube roots and square roots?

Tip: When working with radicals, always check if the expression can be simplified using known identities or special factorizations before approximating.