Express as an equivalent fraction with rational denominator.
`(sqrt(10)+sqrt(5)-sqrt(3))/(sqrt((3))+sqrt((10))-sqrt((5)))`

Answer

To express the given expression \((\sqrt{10} + \sqrt{5} - \sqrt{3}) / (\sqrt{3} + \sqrt{10} - \sqrt{5})\) as an equivalent fraction with a rational denominator, we will follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{\sqrt{10} + \sqrt{5} - \sqrt{3}}{\sqrt{3} + \sqrt{10} - \sqrt{5}} \]

### Step 2: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \((\sqrt{3} + \sqrt{10} - \sqrt{5})\) is \((\sqrt{3} + \sqrt{10} + \sqrt{5})\).

So, we multiply:
\[ \frac{(\sqrt{10} + \sqrt{5} - \sqrt{3})(\sqrt{3} + \sqrt{10} + \sqrt{5})}{(\sqrt{3} + \sqrt{10} - \sqrt{5})(\sqrt{3} + \sqrt{10} + \sqrt{5})} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:
\[ (\sqrt{3} + \sqrt{10})^2 - (\sqrt{5})^2 = (3 + 10 + 2\sqrt{30}) - 5 = 8 + 2\sqrt{30} \]

### Step 4: Expand the numerator
Now we expand the numerator:
\[ (\sqrt{10} + \sqrt{5} - \sqrt{3})(\sqrt{3} + \sqrt{10} + \sqrt{5}) = \sqrt{10}\sqrt{3} + \sqrt{10}\sqrt{10} + \sqrt{10}\sqrt{5} + \sqrt{5}\sqrt{3} + \sqrt{5}\sqrt{10} + \sqrt{5}\sqrt{5} - \sqrt{3}\sqrt{3} - \sqrt{3}\sqrt{10} - \sqrt{3}\sqrt{5} \]
This simplifies to:
\[ \sqrt{30} + 10 + \sqrt{50} + \sqrt{15} + \sqrt{50} + 5 - 3 - \sqrt{30} - \sqrt{15} - \sqrt{15} \]
Combining like terms gives:
\[ 10 + 5 - 3 + 2\sqrt{50} + \sqrt{30} - 3\sqrt{15} \]
This simplifies to:
\[ 12 + 2\sqrt{50} - 3\sqrt{15} \]

### Step 5: Write the final expression
Now we can write the final expression:
\[ \frac{12 + 2\sqrt{50} - 3\sqrt{15}}{8 + 2\sqrt{30}} \]

### Step 6: Further simplify if possible
We can factor out a 2 from the denominator:
\[ = \frac{12 + 2\sqrt{50} - 3\sqrt{15}}{2(4 + \sqrt{30})} = \frac{6 + \sqrt{50} - \frac{3}{2}\sqrt{15}}{4 + \sqrt{30}} \]

### Final Answer:
Thus, the equivalent fraction with a rational denominator is:
\[ \frac{6 + \sqrt{50} - \frac{3}{2}\sqrt{15}}{4 + \sqrt{30}} \]