If `sqrt(15-xsqrt(14))=sqrt(8)-sqrt(7)` , then find the value of `x` .

Answer

To solve the equation \( \sqrt{15 - x\sqrt{14}} = \sqrt{8} - \sqrt{7} \), we will follow these steps:

### Step 1: Square both sides
To eliminate the square root, we square both sides of the equation:

\[ \left(\sqrt{15 - x\sqrt{14}}\right)^2 = \left(\sqrt{8} - \sqrt{7}\right)^2 \]

This simplifies to:

\[ 15 - x\sqrt{14} = (\sqrt{8})^2 - 2\sqrt{8}\sqrt{7} + (\sqrt{7})^2 \]

### Step 2: Simplify the right side
Calculating the squares and the product on the right side:

\[ 15 - x\sqrt{14} = 8 - 2\sqrt{56} + 7 \]

Combine the constants:

\[ 15 - x\sqrt{14} = 15 - 2\sqrt{56} \]

### Step 3: Set the equations equal
Now we can set the two sides equal to each other:

\[ 15 - x\sqrt{14} = 15 - 2\sqrt{56} \]

### Step 4: Cancel out the constants
Subtract 15 from both sides:

\[ -x\sqrt{14} = -2\sqrt{56} \]

### Step 5: Remove the negative sign
Multiplying both sides by -1 gives:

\[ x\sqrt{14} = 2\sqrt{56} \]

### Step 6: Simplify \(\sqrt{56}\)
We can simplify \(\sqrt{56}\):

\[ \sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14} \]

### Step 7: Substitute back
Now substitute this back into the equation:

\[ x\sqrt{14} = 2 \cdot 2\sqrt{14} \]

This simplifies to:

\[ x\sqrt{14} = 4\sqrt{14} \]

### Step 8: Divide by \(\sqrt{14}\)
Now, divide both sides by \(\sqrt{14}\):

\[ x = 4 \]

### Final Answer
Thus, the value of \(x\) is:

\[ \boxed{4} \]