If `sqrt(17+ xsqrt(11)) = sqrt(11) + sqrt(6)`, then value of `x^(2)` will be:

Options

A. `sqrt(11)`
B. 23
C. `sqrt(6)`
D. 24

Answer

To solve the equation \( \sqrt{17 + x\sqrt{11}} = \sqrt{11} + \sqrt{6} \), we will follow these steps:

### Step 1: Square both sides
We start by squaring both sides of the equation to eliminate the square roots.

\[ \left(\sqrt{17 + x\sqrt{11}}\right)^2 = \left(\sqrt{11} + \sqrt{6}\right)^2 \]

### Step 2: Expand both sides
Now, we expand both sides of the equation.

Left-hand side:
\[ 17 + x\sqrt{11} \]

Right-hand side (using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \)):
\[ (\sqrt{11})^2 + 2(\sqrt{11})(\sqrt{6}) + (\sqrt{6})^2 = 11 + 2\sqrt{66} + 6 = 17 + 2\sqrt{66} \]

### Step 3: Set the equations equal
Now we set the left-hand side equal to the right-hand side:

\[ 17 + x\sqrt{11} = 17 + 2\sqrt{66} \]

### Step 4: Simplify the equation
We can subtract 17 from both sides:

\[ x\sqrt{11} = 2\sqrt{66} \]

### Step 5: Solve for \( x \)
Now, we can isolate \( x \) by dividing both sides by \( \sqrt{11} \):

\[ x = \frac{2\sqrt{66}}{\sqrt{11}} \]

### Step 6: Simplify \( x \)
We can simplify \( \frac{2\sqrt{66}}{\sqrt{11}} \):

\[ x = 2\sqrt{\frac{66}{11}} = 2\sqrt{6} \]

### Step 7: Find \( x^2 \)
Now, we need to find \( x^2 \):

\[ x^2 = (2\sqrt{6})^2 = 4 \cdot 6 = 24 \]

### Final Answer
Thus, the value of \( x^2 \) is \( \boxed{24} \).

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