If $$\left(x + 6\sqrt{2}\right)^{\frac{1}{2}} - \left(x - 6\sqrt{2}\right)^{\frac{1}{2}} = 2\sqrt{2},$$

then $$x$$ equals

Question

If  $$\left(x + 6\sqrt{2}\right)^{\frac{1}{2}} - \left(x - 6\sqrt{2}\right)^{\frac{1}{2}} = 2\sqrt{2},$$

then $$x$$ equals

Accepted Answer

The solution to the equation  
$$
\sqrt{x + 6\sqrt{2}} - \sqrt{x - 6\sqrt{2}} = 2\sqrt{2}
$$  
is as follows:

### Step 1: Square both sides
Squaring both sides of the equation, we get:  
$$
x + 6\sqrt{2} + x - 6\sqrt{2} - 2\sqrt{x^2 - 72} = 8.
$$  
Simplify:  
$$
2x - 2\sqrt{x^2 - 72} = 8.
$$

### Step 2: Rearrange terms
Rearranging, we obtain:  
$$
x - \sqrt{x^2 - 72} = 4.
$$

### Step 3: Square again
Square both sides again:  
$$
(x - \sqrt{x^2 - 72})^2 = 4^2.
$$  
Expand:  
$$
x^2 - 2x\sqrt{x^2 - 72} + (x^2 - 72) = 16.
$$  
Simplify:  
$$
2x^2 - 72 - 2x\sqrt{x^2 - 72} = 16.
$$

### Step 4: Solve for $$x$$
From the attached image, solving the simplified equation yields:  
$$
x = 11.
$$

### Final Answer:
The value of $$x$$ is **11**.