The value of `sqrt(9-2sqrt(11-6sqrt(2)))` is closest to :

Options

A. `2.7`
B. `2.9`
C. `2.4`
D. `2.1`

Answer

To solve the expression \( \sqrt{9 - 2\sqrt{11 - 6\sqrt{2}}} \), we will break it down step by step.

### Step 1: Simplify the inner square root
We start with the expression inside the square root:
\[ 11 - 6\sqrt{2} \]

We can try to express \( 11 - 6\sqrt{2} \) in the form of \( (a - b\sqrt{c})^2 \).

### Step 2: Identify \( a \) and \( b \)
Let’s assume:
\[ 11 - 6\sqrt{2} = (a - b\sqrt{2})^2 \]
Expanding the right side, we get:
\[ a^2 + 2ab\sqrt{2} + 2b^2 \]
This means:
- \( a^2 + 2b^2 = 11 \) (1)
- \( 2ab = -6 \) (2)

From equation (2), we can express \( ab \):
\[ ab = -3 \]
Thus, \( b = -\frac{3}{a} \).

### Step 3: Substitute \( b \) in equation (1)
Substituting \( b \) in equation (1):
\[ a^2 + 2\left(-\frac{3}{a}\right)^2 = 11 \]
\[ a^2 + 2\left(\frac{9}{a^2}\right) = 11 \]
Multiplying through by \( a^2 \) to eliminate the fraction:
\[ a^4 - 11a^2 + 18 = 0 \]

### Step 4: Let \( x = a^2 \)
Let \( x = a^2 \):
\[ x^2 - 11x + 18 = 0 \]
Factoring this quadratic:
\[ (x - 2)(x - 9) = 0 \]
Thus, \( x = 2 \) or \( x = 9 \).

### Step 5: Find \( a \) and \( b \)
If \( x = 2 \):
\[ a^2 = 2 \Rightarrow a = \sqrt{2} \]
Then from \( ab = -3 \):
\[ b = -\frac{3}{\sqrt{2}} \]

If \( x = 9 \):
\[ a^2 = 9 \Rightarrow a = 3 \]
Then from \( ab = -3 \):
\[ b = -1 \]

### Step 6: Choose the correct values
We can use \( a = 3 \) and \( b = -1 \) because it gives us:
\[ 11 - 6\sqrt{2} = (3 - \sqrt{2})^2 \]

### Step 7: Substitute back into the original expression
Now substituting back:
\[ \sqrt{9 - 2\sqrt{11 - 6\sqrt{2}}} = \sqrt{9 - 2(3 - \sqrt{2})} \]
\[ = \sqrt{9 - 6 + 2\sqrt{2}} \]
\[ = \sqrt{3 + 2\sqrt{2}} \]

### Step 8: Simplify \( \sqrt{3 + 2\sqrt{2}} \)
We can express \( 3 + 2\sqrt{2} \) as \( (a + b\sqrt{2})^2 \):
Let’s assume:
\[ 3 + 2\sqrt{2} = (a + b\sqrt{2})^2 \]
Expanding gives:
\[ a^2 + 2b^2 + 2ab\sqrt{2} \]
This means:
- \( a^2 + 2b^2 = 3 \) (3)
- \( 2ab = 2 \) (4)

From equation (4):
\[ ab = 1 \Rightarrow b = \frac{1}{a} \]

### Step 9: Substitute \( b \) in equation (3)
Substituting \( b \) in equation (3):
\[ a^2 + 2\left(\frac{1}{a}\right)^2 = 3 \]
Multiplying through by \( a^2 \):
\[ a^4 - 3a^2 + 2 = 0 \]
Let \( y = a^2 \):
\[ y^2 - 3y + 2 = 0 \]
Factoring gives:
\[ (y - 1)(y - 2) = 0 \]
Thus, \( y = 1 \) or \( y = 2 \).

### Step 10: Find \( a \) and \( b \)
If \( y = 1 \):
\[ a^2 = 1 \Rightarrow a = 1 \]
Then \( b = 1 \).

If \( y = 2 \):
\[ a^2 = 2 \Rightarrow a = \sqrt{2} \]
Then \( b = \frac{1}{\sqrt{2}} \).

### Final Step: Evaluate the expression
Using \( a = 1 \) and \( b = 1 \):
\[ \sqrt{3 + 2\sqrt{2}} = 1 + \sqrt{2} \]
Since \( \sqrt{2} \approx 1.414 \):
\[ 1 + \sqrt{2} \approx 1 + 1.414 = 2.414 \]

Thus, the value of \( \sqrt{9 - 2\sqrt{11 - 6\sqrt{2}}} \) is closest to **2.4**.