`sqrt(6 + sqrt(6 + sqrt(6 ) ) )` ?

Options

A. 2.3
B. 3
C. 6
D. 6.3

Answer

To solve the expression \( \sqrt{6 + \sqrt{6 + \sqrt{6}}} \), we can follow these steps:

### Step 1: Define the Expression
Let \( x = \sqrt{6 + \sqrt{6 + \sqrt{6}}} \).

### Step 2: Square Both Sides
To eliminate the outer square root, we square both sides:
\[ x^2 = 6 + \sqrt{6 + \sqrt{6}} \]

### Step 3: Isolate the Inner Square Root
Next, we isolate the inner square root:
\[ \sqrt{6 + \sqrt{6}} = x^2 - 6 \]

### Step 4: Square Again
Now, we square both sides again to eliminate the square root:
\[ 6 + \sqrt{6} = (x^2 - 6)^2 \]

### Step 5: Expand the Right Side
Expanding the right side:
\[ 6 + \sqrt{6} = x^4 - 12x^2 + 36 \]

### Step 6: Rearrange the Equation
Rearranging gives us:
\[ x^4 - 12x^2 + 30 - \sqrt{6} = 0 \]

### Step 7: Approximate Values
To find \( x \), we can estimate \( \sqrt{6} \approx 2.45 \), so we can simplify our calculations.

### Step 8: Estimate \( x \)
We can try some values for \( x \):
- If \( x = 3 \):
\[ 3^2 = 9 \Rightarrow 6 + \sqrt{6 + \sqrt{6}} = 9 \Rightarrow \sqrt{6 + \sqrt{6}} = 3 \]
This means:
\[ 6 + \sqrt{6} = 9 \Rightarrow \sqrt{6} = 3 \]
This is not true, but we can check if \( x = 3 \) works in the original equation.

### Step 9: Check \( x = 3 \)
Substituting \( x = 3 \) back into the original equation:
\[ \sqrt{6 + \sqrt{6 + \sqrt{6}}} = 3 \]
Calculating:
\[ \sqrt{6 + \sqrt{6 + \sqrt{6}}} = \sqrt{6 + \sqrt{6 + 2.45}} \approx \sqrt{6 + 3} = \sqrt{9} = 3 \]

### Conclusion
Thus, the value of \( \sqrt{6 + \sqrt{6 + \sqrt{6}}} \) is \( 3 \).