The square root of `33 - 4 sqrt35` is :

Options

A. `pm (2 sqrt7 + sqrt5)`
B. `pm (sqrt7 + 2 sqrt5)`
C. `+(sqrt7 - 2sqrt5)`
D. `pm (2sqrt7 - sqrt 5)`

Answer

To find the square root of \( 33 - 4\sqrt{35} \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ 33 - 4\sqrt{35} \]
We can express \( 33 \) in a different form to facilitate the square root calculation.

### Step 2: Identify a Suitable Form
Notice that:
\[ 4\sqrt{35} = 4 \cdot \sqrt{7 \cdot 5} = 2 \cdot 2 \cdot \sqrt{7} \cdot \sqrt{5} \]
We can rewrite \( 33 \) as:
\[ 33 = (2\sqrt{7})^2 + (\sqrt{5})^2 \]
Thus, we can express \( 33 - 4\sqrt{35} \) as:
\[ (2\sqrt{7})^2 + (\sqrt{5})^2 - 2 \cdot (2\sqrt{7}) \cdot (\sqrt{5}) \]

### Step 3: Recognize the Perfect Square
The expression now resembles the expansion of a perfect square:
\[ a^2 + b^2 - 2ab = (a - b)^2 \]
where \( a = 2\sqrt{7} \) and \( b = \sqrt{5} \). Therefore, we can write:
\[ 33 - 4\sqrt{35} = (2\sqrt{7} - \sqrt{5})^2 \]

### Step 4: Take the Square Root
Now, we take the square root of both sides:
\[ \sqrt{33 - 4\sqrt{35}} = \sqrt{(2\sqrt{7} - \sqrt{5})^2} \]
This simplifies to:
\[ 2\sqrt{7} - \sqrt{5} \]
However, we must remember that taking the square root can yield both positive and negative values:
\[ \sqrt{33 - 4\sqrt{35}} = \pm (2\sqrt{7} - \sqrt{5}) \]

### Final Answer
Thus, the square root of \( 33 - 4\sqrt{35} \) is:
\[ 2\sqrt{7} - \sqrt{5} \quad \text{or} \quad -(2\sqrt{7} - \sqrt{5}) \]