The value of : `sqrt(-sqrt(3)+sqrt(3+8sqrt(7+4sqrt(3))))` is

Options

A. `1`
B. `2`
C. `3`
D. `8`

Answer

To solve the expression \( \sqrt{-\sqrt{3} + \sqrt{3 + 8\sqrt{7 + 4\sqrt{3}}}} \), we will break it down step by step.

### Step 1: Simplify the inner expression \( \sqrt{7 + 4\sqrt{3}} \)

To simplify \( \sqrt{7 + 4\sqrt{3}} \), we can express it in the form \( \sqrt{a} + \sqrt{b} \).

Assume:
\[ \sqrt{7 + 4\sqrt{3}} = \sqrt{a} + \sqrt{b} \]
Squaring both sides gives:
\[ 7 + 4\sqrt{3} = a + b + 2\sqrt{ab} \]
From this, we can equate the rational and irrational parts:
1. \( a + b = 7 \)
2. \( 2\sqrt{ab} = 4\sqrt{3} \) → \( \sqrt{ab} = 2\sqrt{3} \) → \( ab = 12 \)

Now we have the system of equations:
1. \( a + b = 7 \)
2. \( ab = 12 \)

### Step 2: Solve for \( a \) and \( b \)

The values of \( a \) and \( b \) can be found using the quadratic equation:
\[ x^2 - (a+b)x + ab = 0 \implies x^2 - 7x + 12 = 0 \]
Using the quadratic formula:
\[ x = \frac{7 \pm \sqrt{7^2 - 4 \cdot 12}}{2} = \frac{7 \pm \sqrt{49 - 48}}{2} = \frac{7 \pm 1}{2} \]
This gives:
\[ x = 4 \quad \text{or} \quad x = 3 \]
Thus, \( a = 4 \) and \( b = 3 \) (or vice versa).

### Step 3: Substitute back to find \( \sqrt{7 + 4\sqrt{3}} \)

Now we have:
\[ \sqrt{7 + 4\sqrt{3}} = \sqrt{4} + \sqrt{3} = 2 + \sqrt{3} \]

### Step 4: Substitute back into the original expression

Now substitute \( \sqrt{7 + 4\sqrt{3}} \) back into the expression:
\[ \sqrt{3 + 8\sqrt{7 + 4\sqrt{3}}} = \sqrt{3 + 8(2 + \sqrt{3})} \]
Calculating this gives:
\[ = \sqrt{3 + 16 + 8\sqrt{3}} = \sqrt{19 + 8\sqrt{3}} \]

### Step 5: Simplify \( \sqrt{19 + 8\sqrt{3}} \)

Assume:
\[ \sqrt{19 + 8\sqrt{3}} = \sqrt{a} + \sqrt{b} \]
Squaring both sides gives:
\[ 19 + 8\sqrt{3} = a + b + 2\sqrt{ab} \]
From this, we equate:
1. \( a + b = 19 \)
2. \( 2\sqrt{ab} = 8\sqrt{3} \) → \( \sqrt{ab} = 4\sqrt{3} \) → \( ab = 48 \)

### Step 6: Solve for \( a \) and \( b \) again

Using the quadratic equation:
\[ x^2 - 19x + 48 = 0 \]
Using the quadratic formula:
\[ x = \frac{19 \pm \sqrt{19^2 - 4 \cdot 48}}{2} = \frac{19 \pm \sqrt{361 - 192}}{2} = \frac{19 \pm \sqrt{169}}{2} = \frac{19 \pm 13}{2} \]
This gives:
\[ x = 16 \quad \text{or} \quad x = 3 \]
Thus, \( a = 16 \) and \( b = 3 \) (or vice versa).

### Step 7: Substitute back to find \( \sqrt{19 + 8\sqrt{3}} \)

Now we have:
\[ \sqrt{19 + 8\sqrt{3}} = \sqrt{16} + \sqrt{3} = 4 + \sqrt{3} \]

### Step 8: Substitute into the original expression

Now substitute back into the original expression:
\[ \sqrt{-\sqrt{3} + (4 + \sqrt{3})} = \sqrt{4} = 2 \]

### Final Result

Thus, the value of the expression is:
\[ \boxed{2} \]