Find the square root of : `21-4sqrt(5)+8sqrt(3)-4sqrt(15)`.

Answer

To find the square root of the expression \( 21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15} \), we can assume it can be expressed in the form \( \sqrt{a} + \sqrt{b} \) and then square it to match the given expression.

### Step 1: Assume the form of the square root
Let us assume:
\[ \sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}} = \sqrt{a} + \sqrt{b} \]

### Step 2: Square both sides
Squaring both sides gives:
\[ 21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15} = a + b + 2\sqrt{ab} \]

### Step 3: Identify \( a \) and \( b \)
From the above equation, we can separate the rational and irrational parts. Thus, we have:
\[ a + b = 21 \]
\[ 2\sqrt{ab} = -4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15} \]

### Step 4: Find \( ab \)
To find \( ab \), we can rewrite the equation for the irrational parts:
\[ 2\sqrt{ab} = -4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15} \]
Dividing by 2:
\[ \sqrt{ab} = -2\sqrt{5} + 4\sqrt{3} - 2\sqrt{15} \]
Squaring both sides gives:
\[ ab = (-2\sqrt{5} + 4\sqrt{3} - 2\sqrt{15})^2 \]

### Step 5: Expand \( ab \)
Expanding this expression:
\[ ab = 4 \cdot 5 + 16 \cdot 3 + 4 \cdot 15 - 2 \cdot 2 \cdot 4\sqrt{3} \cdot 2\sqrt{5} + 2 \cdot 2 \cdot 4\sqrt{3} \cdot 2\sqrt{15} - 2 \cdot 2 \cdot 5 \]
Calculating each term:
\[ = 20 + 48 + 60 - 16\sqrt{15} + 16\sqrt{10} - 20 \]
This simplifies to:
\[ = 108 - 16\sqrt{15} + 16\sqrt{10} \]

### Step 6: Solve for \( a \) and \( b \)
Now we have the two equations:
1. \( a + b = 21 \)
2. \( ab = 108 - 16\sqrt{15} + 16\sqrt{10} \)

Using these equations, we can find \( a \) and \( b \) by solving the quadratic equation:
\[ x^2 - (a+b)x + ab = 0 \]
This gives:
\[ x^2 - 21x + (108 - 16\sqrt{15} + 16\sqrt{10}) = 0 \]

### Step 7: Find the roots
Using the quadratic formula:
\[ x = \frac{21 \pm \sqrt{21^2 - 4(108 - 16\sqrt{15} + 16\sqrt{10})}}{2} \]

### Step 8: Final expression
After solving for \( a \) and \( b \), we can express the square root:
\[ \sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}} = \sqrt{a} + \sqrt{b} \]
The final answer will be:
\[ \sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}} = 2\sqrt{3} + 2 - \sqrt{5} \]