`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 simplify it step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{12}} \]
We can rewrite \( 21 \) as \( 20 + 1 \) and \( 4\sqrt{12} \) as \( 4 \cdot 2\sqrt{3} = 8\sqrt{3} \). Thus, we have:
\[ \sqrt{(20 + 1) - 4\sqrt{5} + 8\sqrt{3} - 8\sqrt{3}} \]
This simplifies to:
\[ \sqrt{21 - 4\sqrt{5}} \]

### Step 2: Combine like terms
Notice that \( 8\sqrt{3} - 8\sqrt{3} = 0 \), so we can ignore that part. The expression now looks like:
\[ \sqrt{21 - 4\sqrt{5}} \]

### Step 3: Express \( 21 - 4\sqrt{5} \) in a different form
We want to express \( 21 - 4\sqrt{5} \) in the form of \( (a - b)^2 \). Let's assume:
\[ 21 - 4\sqrt{5} = (x - y)^2 \]
Expanding \( (x - y)^2 \) gives us:
\[ x^2 - 2xy + y^2 \]
We need to find \( x \) and \( y \) such that:
- \( x^2 + y^2 = 21 \)
- \( 2xy = 4\sqrt{5} \)

### Step 4: Solve for \( x \) and \( y \)
From \( 2xy = 4\sqrt{5} \), we can simplify to:
\[ xy = 2\sqrt{5} \]
Now we have two equations:
1. \( x^2 + y^2 = 21 \)
2. \( xy = 2\sqrt{5} \)

Using the identity \( (x + y)^2 = x^2 + y^2 + 2xy \), we can express \( x + y \):
\[ (x + y)^2 = 21 + 4\sqrt{5} \]

### Step 5: Find \( x \) and \( y \)
Let's assume \( x = \sqrt{5} + 4 \) and \( y = \sqrt{5} - 1 \). Then:
\[ x^2 + y^2 = (\sqrt{5} + 4)^2 + (\sqrt{5} - 1)^2 \]
Calculating:
\[ = (5 + 8\sqrt{5} + 16) + (5 - 2\sqrt{5} + 1) = 21 + 6\sqrt{5} \]

### Step 6: Final simplification
Now we can express:
\[ \sqrt{21 - 4\sqrt{5}} = \sqrt{(\sqrt{5} + 4 - (\sqrt{5} - 1))^2} = \sqrt{(5 - 1)^2} = \sqrt{4^2} = 4 \]

Thus, the final answer is:
\[ \sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{12}} = \sqrt{20 - 1} = 4 \]