What is the value of `[(12)/((sqrt(5)+sqrt(3)))+(18)/((sqrt(5)-sqrt(3)))]`?

Options

A. `15(sqrt(15)-sqrt(3))`
B. `3 ( 5 sqrt(5)+sqrt(3))`
C. `15 ( sqrt(5) + sqrt(3))`
D. `3 ( 3 sqrt(5) + sqrt(3))`

Answer

To solve the expression \(\left[\frac{12}{\sqrt{5} + \sqrt{3}} + \frac{18}{\sqrt{5} - \sqrt{3}}\right]\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})\).

### Step 2: Rewrite the fractions
Rewriting the fractions with the common denominator, we have:
\[ \frac{12(\sqrt{5} - \sqrt{3}) + 18(\sqrt{5} + \sqrt{3})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})} \]

### Step 3: Simplify the numerator
Now, we simplify the numerator:
\[ 12(\sqrt{5} - \sqrt{3}) + 18(\sqrt{5} + \sqrt{3}) = 12\sqrt{5} - 12\sqrt{3} + 18\sqrt{5} + 18\sqrt{3} \]
Combine like terms:
\[ (12\sqrt{5} + 18\sqrt{5}) + (-12\sqrt{3} + 18\sqrt{3}) = 30\sqrt{5} + 6\sqrt{3} \]

### Step 4: Simplify the denominator
The denominator can be simplified using the difference of squares:
\[ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \]

### Step 5: Combine the results
Now, we can write the entire expression as:
\[ \frac{30\sqrt{5} + 6\sqrt{3}}{2} \]

### Step 6: Factor out the common term
We can factor out a 6 from the numerator:
\[ \frac{6(5\sqrt{5} + \sqrt{3})}{2} = 3(5\sqrt{5} + \sqrt{3}) \]

### Final Result
Thus, the value of the expression is:
\[ 3(5\sqrt{5} + \sqrt{3}) \]