Find the square root of `21 - 4sqrt5 + 8sqrt3 - 4 sqrt15`

Options

A. `(2sqrt3 + 2 - sqrt5)`
B. `(2sqrt3 - 2 - sqrt5)`
C. `(2sqrt3 + 2+ sqrt5)`
D. `(2+ sqrt5-2sqrt3)`

Answer

To find the square root of the expression \( 21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15} \]

### Step 2: Identify a possible square
We want to express this in the form of \( (a + b + c)^2 \). The expansion of \( (a + b + c)^2 \) is given by:
\[ a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \]
We will try to identify \( a \), \( b \), and \( c \) such that their squares and the cross products match our expression.

### Step 3: Assume values for \( a \), \( b \), and \( c \)
Let’s assume:
- \( a = 2\sqrt{3} \)
- \( b = -2 \)
- \( c = -\sqrt{5} \)

### Step 4: Calculate \( a^2 \), \( b^2 \), and \( c^2 \)
Now we calculate:
\[ a^2 = (2\sqrt{3})^2 = 4 \cdot 3 = 12 \]
\[ b^2 = (-2)^2 = 4 \]
\[ c^2 = (-\sqrt{5})^2 = 5 \]

### Step 5: Calculate the sum of squares
Now, we sum these squares:
\[ a^2 + b^2 + c^2 = 12 + 4 + 5 = 21 \]

### Step 6: Calculate the cross products
Next, we calculate the cross products:
\[ 2ab = 2 \cdot (2\sqrt{3}) \cdot (-2) = -8\sqrt{3} \]
\[ 2bc = 2 \cdot (-2) \cdot (-\sqrt{5}) = 4\sqrt{5} \]
\[ 2ca = 2 \cdot (-\sqrt{5}) \cdot (2\sqrt{3}) = -4\sqrt{15} \]

### Step 7: Combine all terms
Now, we combine these results:
\[ a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = 21 - 8\sqrt{3} + 4\sqrt{5} - 4\sqrt{15} \]

### Step 8: Verify the expression
We see that the expression matches:
\[ 21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15} \]

### Step 9: Take the square root
Thus, we can conclude:
\[ \sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}} = 2\sqrt{3} - 2 - \sqrt{5} \]

### Final Answer
The final answer is:
\[ 2\sqrt{3} - 2 - \sqrt{5} \]