Value of `sqrt(4 + sqrt(15))` is

Options

A. `(sqrt(5) - sqrt(3))/(2)`
B. `(sqrt(5) - sqrt(3))/(sqrt2)`
C. `(sqrt(5) + sqrt(3))/(sqrt2)`
D. `(sqrt(5) + sqrt(3))/(2)`

Answer

To find the value of \( \sqrt{4 + \sqrt{15}} \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{4 + \sqrt{15}} \]

### Step 2: Multiply and divide by 2
To simplify the expression, we can multiply and divide by 2:
\[ \sqrt{4 + \sqrt{15}} = \sqrt{2 \cdot \frac{4 + \sqrt{15}}{2}} = \sqrt{2 \cdot (2 + \frac{\sqrt{15}}{2})} \]

### Step 3: Simplify the expression inside the square root
Now, we can rewrite \( 4 + \sqrt{15} \) as:
\[ \sqrt{2 \cdot (2 + \frac{\sqrt{15}}{2})} \]

### Step 4: Express in terms of squares
Next, we can express \( 4 + \sqrt{15} \) in a form that resembles a perfect square:
\[ 4 + \sqrt{15} = \left(\sqrt{5} + \sqrt{3}\right)^2 \]
This is because:
\[ (\sqrt{5} + \sqrt{3})^2 = 5 + 3 + 2\sqrt{15} = 8 + 2\sqrt{15} \]

### Step 5: Substitute back into the square root
Now substituting back, we have:
\[ \sqrt{4 + \sqrt{15}} = \sqrt{\left(\sqrt{5} + \sqrt{3}\right)^2} \]

### Step 6: Simplify the square root
Taking the square root gives us:
\[ \sqrt{4 + \sqrt{15}} = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{2}} \]

### Final Result
Thus, the value of \( \sqrt{4 + \sqrt{15}} \) is:
\[ \frac{\sqrt{5} + \sqrt{3}}{\sqrt{2}} \]

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