Simplify :
` ( 5- sqrt(10 ))/( 5+ sqrt( 10 ) ) -( 5+ sqrt(10 ))/( 5-sqrt(10))`

Answer

To simplify the expression \((5 - \sqrt{10})/(5 + \sqrt{10}) - (5 + \sqrt{10})/(5 - \sqrt{10})\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((5 + \sqrt{10})(5 - \sqrt{10})\).

### Step 2: Rewrite the fractions
We can rewrite the expression as:
\[ \frac{(5 - \sqrt{10})^2 - (5 + \sqrt{10})^2}{(5 + \sqrt{10})(5 - \sqrt{10})} \]

### Step 3: Expand the numerators
Using the identity \(a^2 - b^2 = (a - b)(a + b)\), we can expand the numerators:
- For \((5 - \sqrt{10})^2\):
\[ (5 - \sqrt{10})^2 = 5^2 - 2 \cdot 5 \cdot \sqrt{10} + (\sqrt{10})^2 = 25 - 10\sqrt{10} + 10 = 35 - 10\sqrt{10} \]
- For \((5 + \sqrt{10})^2\):
\[ (5 + \sqrt{10})^2 = 5^2 + 2 \cdot 5 \cdot \sqrt{10} + (\sqrt{10})^2 = 25 + 10\sqrt{10} + 10 = 35 + 10\sqrt{10} \]

### Step 4: Substitute back into the expression
Now substituting back, we have:
\[ \frac{(35 - 10\sqrt{10}) - (35 + 10\sqrt{10})}{(5 + \sqrt{10})(5 - \sqrt{10})} \]

### Step 5: Simplify the numerator
The numerator simplifies to:
\[ 35 - 10\sqrt{10} - 35 - 10\sqrt{10} = -20\sqrt{10} \]

### Step 6: Simplify the denominator
The denominator simplifies as follows:
\[ (5 + \sqrt{10})(5 - \sqrt{10}) = 5^2 - (\sqrt{10})^2 = 25 - 10 = 15 \]

### Step 7: Combine the results
Now we can combine the results:
\[ \frac{-20\sqrt{10}}{15} \]

### Step 8: Simplify the fraction
This can be simplified by dividing both the numerator and the denominator by 5:
\[ \frac{-4\sqrt{10}}{3} \]

### Final Answer
Thus, the simplified expression is:
\[ -\frac{4\sqrt{10}}{3} \]