### Question Statement:
Find the value of the expression:
$$
\sqrt{4 + \sqrt{15}} + \sqrt{4 - \sqrt{15}} - \sqrt{12 - 4\sqrt{5}}
$$
and determine its nature.

Question

### Question Statement:
Find the value of the expression:  
$$
\sqrt{4 + \sqrt{15}} + \sqrt{4 - \sqrt{15}} - \sqrt{12 - 4\sqrt{5}}
$$  
and determine its nature.

Accepted Answer

To evaluate the given expression:

1. **Simplify $$\sqrt{4 + \sqrt{15}}$$:**  
   Using the identity for perfect squares,  
   $$
   \sqrt{4 + \sqrt{15}} = \sqrt{\frac{5}{2} + \frac{\sqrt{15}}{2}} = \frac{\sqrt{5}}{\sqrt{2}} + \frac{\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{2}}.
   $$

2. **Simplify $$\sqrt{4 - \sqrt{15}}$$:**  
   Similarly,  
   $$
   \sqrt{4 - \sqrt{15}} = \frac{\sqrt{5}}{\sqrt{2}} - \frac{\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{2}}.
   $$

3. **Simplify $$\sqrt{12 - 4\sqrt{5}}$$:**  
   Rewrite $$12 - 4\sqrt{5}$$ as a perfect square:  
   $$
   12 - 4\sqrt{5} = (\sqrt{10} - \sqrt{2})^2.
   $$  
   Thus,  
   $$
   \sqrt{12 - 4\sqrt{5}} = (\sqrt{10} - \sqrt{2}).
   $$

4. **Combine the terms:**  
   Substituting back into the expression:  
   $$
   (\frac{\sqrt{5} + \sqrt{3}}{\sqrt{2}}) + (\frac{\sqrt{5} - \sqrt{3}}{\sqrt{2}}) - (\sqrt{10} - \sqrt{2}),
   $$  
   simplifies to:  
   $$
   2\frac{\sqrt{5}}{\sqrt{2}} - (\sqrt{10} - \sqrt{2}) = 2\frac{\sqrt{5}}{\sqrt{2}} - (\sqrt{10}) + (\sqrt{2}),
   $$  
   which is an irrational number.

Thus, the value of the expression is an **irrational number**.

Answer

Answer

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