Express as an equivalent fraction with rational denominator.
`(sqrt(2)(sqrt(3)+1)(2-sqrt(3)))/([sqrt((2))-1][3sqrt((3))-5][2+sqrt((2))])`

Answer

To express the given expression as an equivalent fraction with a rational denominator, we will follow these steps:

Given expression:
\[ \frac{\sqrt{2}(\sqrt{3}+1)(2-\sqrt{3})}{(\sqrt{2}-1)(3\sqrt{3}-5)(2+\sqrt{2})} \]

### Step 1: Simplify the numerator
First, we will simplify the numerator:
\[ \sqrt{2}(\sqrt{3}+1)(2-\sqrt{3}) \]
We can expand this expression step by step.

1. Expand \((\sqrt{3}+1)(2-\sqrt{3})\):
\[ = \sqrt{3} \cdot 2 - \sqrt{3} \cdot \sqrt{3} + 1 \cdot 2 - 1 \cdot \sqrt{3} \]
\[ = 2\sqrt{3} - 3 + 2 - \sqrt{3} \]
\[ = (2\sqrt{3} - \sqrt{3}) + (2 - 3) = \sqrt{3} - 1 \]

2. Now multiply by \(\sqrt{2}\):
\[ \sqrt{2}(\sqrt{3} - 1) = \sqrt{2}\sqrt{3} - \sqrt{2} = \sqrt{6} - \sqrt{2} \]

So, the numerator simplifies to:
\[ \sqrt{6} - \sqrt{2} \]

### Step 2: Simplify the denominator
Now, let's simplify the denominator:
\[ (\sqrt{2}-1)(3\sqrt{3}-5)(2+\sqrt{2}) \]

1. First, we will multiply \((\sqrt{2}-1)(2+\sqrt{2})\):
\[ = \sqrt{2} \cdot 2 + \sqrt{2} \cdot \sqrt{2} - 1 \cdot 2 - 1 \cdot \sqrt{2} \]
\[ = 2\sqrt{2} + 2 - 2 - \sqrt{2} = (2\sqrt{2} - \sqrt{2}) + (2 - 2) = \sqrt{2} \]

2. Now multiply by \((3\sqrt{3}-5)\):
\[ = \sqrt{2}(3\sqrt{3}-5) = 3\sqrt{6} - 5\sqrt{2} \]

So, the denominator simplifies to:
\[ 3\sqrt{6} - 5\sqrt{2} \]

### Step 3: Form the new fraction
Now we have:
\[ \frac{\sqrt{6} - \sqrt{2}}{3\sqrt{6} - 5\sqrt{2}} \]

### Step 4: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ 3\sqrt{6} + 5\sqrt{2} \]

The new fraction becomes:
\[ \frac{(\sqrt{6} - \sqrt{2})(3\sqrt{6} + 5\sqrt{2})}{(3\sqrt{6} - 5\sqrt{2})(3\sqrt{6} + 5\sqrt{2})} \]

### Step 5: Calculate the denominator
Using the difference of squares:
\[ (3\sqrt{6})^2 - (5\sqrt{2})^2 = 54 - 50 = 4 \]

### Step 6: Calculate the numerator
Now, we expand the numerator:
\[ (\sqrt{6} - \sqrt{2})(3\sqrt{6} + 5\sqrt{2}) = \sqrt{6} \cdot 3\sqrt{6} + \sqrt{6} \cdot 5\sqrt{2} - \sqrt{2} \cdot 3\sqrt{6} - \sqrt{2} \cdot 5\sqrt{2} \]
\[ = 18 + 5\sqrt{12} - 3\sqrt{12} - 10 = 8 + 2\sqrt{12} = 8 + 4\sqrt{3} \]

### Step 7: Final expression
Putting it all together, we have:
\[ \frac{8 + 4\sqrt{3}}{4} \]
\[ = 2 + \sqrt{3} \]

### Final Answer
Thus, the expression simplifies to:
\[ 2 + \sqrt{3} \]