`sqrt(21-4sqrt5+8sqrt3-4sqrt12)=`

Options

A. `sqrt5-2+2sqrt3`
B. `-sqrt5-sqrt4-sqrt12`
C. `-sqrt5+sqrt4+sqrt12`
D. `-sqrt5-sqrt4+sqrt12`

Answer

To solve the expression \( \sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{12}} \), we will follow these steps:

### Step 1: Simplify the expression inside the square root
We start with the expression:
\[ 21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{12} \]
First, we simplify \( \sqrt{12} \):
\[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \]
Thus, we can rewrite the expression:
\[ 21 - 4\sqrt{5} + 8\sqrt{3} - 4(2\sqrt{3}) = 21 - 4\sqrt{5} + 8\sqrt{3} - 8\sqrt{3} \]
This simplifies to:
\[ 21 - 4\sqrt{5} \]

### Step 2: Rewrite \( 21 - 4\sqrt{5} \)
Next, we can express \( 21 \) as \( 20 + 1 \):
\[ 21 - 4\sqrt{5} = (20 + 1) - 4\sqrt{5} \]

### Step 3: Recognize the expression as a perfect square
We can rewrite \( 20 \) as \( (2\sqrt{5})^2 \):
\[ 21 - 4\sqrt{5} = (2\sqrt{5})^2 + 1^2 - 2 \cdot 2\sqrt{5} \cdot 1 \]
This is in the form of \( a^2 + b^2 - 2ab \), which can be factored as:
\[ (a - b)^2 \]
Thus:
\[ 21 - 4\sqrt{5} = (2\sqrt{5} - 1)^2 \]

### Step 4: Take the square root
Now we take the square root of the entire expression:
\[ \sqrt{21 - 4\sqrt{5}} = \sqrt{(2\sqrt{5} - 1)^2} \]
This simplifies to:
\[ 2\sqrt{5} - 1 \]

### Final Answer
Thus, the final answer is:
\[ \sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{12}} = 2\sqrt{5} - 1 \]