Express as an equivalent fraction with rational denominator.
`(8sqrt(3)-3sqrt(5))/(9sqrt((3))-4sqrt((5)))`

Answer

To express the given fraction \((8\sqrt{3} - 3\sqrt{5}) / (9\sqrt{3} - 4\sqrt{5})\) as an equivalent fraction with a rational denominator, we will rationalize the denominator. Here are the steps to solve the problem:

### Step 1: Identify the Conjugate
The denominator is \(9\sqrt{3} - 4\sqrt{5}\). The conjugate of this expression is \(9\sqrt{3} + 4\sqrt{5}\).

### Step 2: Multiply Numerator and Denominator by the Conjugate
We multiply both the numerator and the denominator by the conjugate:
\[ \frac{(8\sqrt{3} - 3\sqrt{5})(9\sqrt{3} + 4\sqrt{5})}{(9\sqrt{3} - 4\sqrt{5})(9\sqrt{3} + 4\sqrt{5})} \]

### Step 3: Expand the Numerator
Using the distributive property (FOIL method), we expand the numerator:
\[ (8\sqrt{3} - 3\sqrt{5})(9\sqrt{3} + 4\sqrt{5}) = 8\sqrt{3} \cdot 9\sqrt{3} + 8\sqrt{3} \cdot 4\sqrt{5} - 3\sqrt{5} \cdot 9\sqrt{3} - 3\sqrt{5} \cdot 4\sqrt{5} \]
Calculating each term:
1. \(8\sqrt{3} \cdot 9\sqrt{3} = 72 \cdot 3 = 216\)
2. \(8\sqrt{3} \cdot 4\sqrt{5} = 32\sqrt{15}\)
3. \(-3\sqrt{5} \cdot 9\sqrt{3} = -27\sqrt{15}\)
4. \(-3\sqrt{5} \cdot 4\sqrt{5} = -12 \cdot 5 = -60\)

Combining these, the numerator becomes:
\[ 216 + 32\sqrt{15} - 27\sqrt{15} - 60 = 216 - 60 + (32 - 27)\sqrt{15} = 156 + 5\sqrt{15} \]

### Step 4: Expand the Denominator
Next, we expand the denominator using the difference of squares:
\[ (9\sqrt{3})^2 - (4\sqrt{5})^2 = 81 \cdot 3 - 16 \cdot 5 = 243 - 80 = 163 \]

### Step 5: Combine the Results
Now we can write the equivalent fraction:
\[ \frac{156 + 5\sqrt{15}}{163} \]

### Final Answer
The equivalent fraction with a rational denominator is:
\[ \frac{156 + 5\sqrt{15}}{163} \]