If ` x = 5 - sqrt(21)` , then the value of
`( sqrt(x))/( sqrt(32 - 2 x) - sqrt( 21))` is

Options

A. `(1)/(sqrt(2)) ( sqrt(3) - sqrt(7))`
B. `(1)/( sqrt(2)) ( sqrt(7) - sqrt(3))`
C. `(1)/( sqrt(2)) ( sqrt(7) + sqrt(3))`
D. `(1)/( sqrt(2)) ( 7 - sqrt(3))`

Answer

To solve the problem step by step, we start with the given value of \( x \):

### Step 1: Substitute the value of \( x \)
Given:
\[ x = 5 - \sqrt{21} \]

### Step 2: Calculate \( 2x \)
We need to find \( 2x \):
\[ 2x = 2(5 - \sqrt{21}) = 10 - 2\sqrt{21} \]

### Step 3: Calculate \( \sqrt{x} \)
To find \( \sqrt{x} \), we first express \( x \) in a suitable form:
\[ \sqrt{x} = \sqrt{5 - \sqrt{21}} \]

### Step 4: Rewrite \( \sqrt{x} \)
We can rewrite \( 5 - \sqrt{21} \) in terms of squares:
\[ 5 - \sqrt{21} = \left(\sqrt{7} - \sqrt{3}\right)^2 \]
Thus,
\[ \sqrt{x} = \sqrt{\left(\sqrt{7} - \sqrt{3}\right)^2} = \sqrt{7} - \sqrt{3} \]

### Step 5: Calculate \( \sqrt{32 - 2x} \)
Now we calculate \( \sqrt{32 - 2x} \):
\[ 32 - 2x = 32 - (10 - 2\sqrt{21}) = 32 - 10 + 2\sqrt{21} = 22 + 2\sqrt{21} \]
Next, we can express \( 22 + 2\sqrt{21} \) in a square form:
\[ 22 + 2\sqrt{21} = \left(\sqrt{21} + 1\right)^2 \]
Thus,
\[ \sqrt{32 - 2x} = \sqrt{\left(\sqrt{21} + 1\right)^2} = \sqrt{21} + 1 \]

### Step 6: Substitute into the expression
Now we substitute \( \sqrt{x} \) and \( \sqrt{32 - 2x} \) into the original expression:
\[ \frac{\sqrt{x}}{\sqrt{32 - 2x} - \sqrt{21}} = \frac{\sqrt{7} - \sqrt{3}}{(\sqrt{21} + 1) - \sqrt{21}} = \frac{\sqrt{7} - \sqrt{3}}{1} \]

### Step 7: Final Result
Thus, the value of the expression is:
\[ \sqrt{7} - \sqrt{3} \]

### Step 8: Divide by \( \sqrt{2} \)
To express this in the required form:
\[ \frac{\sqrt{7} - \sqrt{3}}{\sqrt{2}} \]

### Conclusion:
The final answer is:
\[ \frac{\sqrt{7} - \sqrt{3}}{\sqrt{2}} \]