To rationalize the denominator of \( \frac{2}{\sqrt{3} - \sqrt{5}} \), follow these steps:
### Step 1: Identify the Conjugate
The first step is to identify the conjugate of the denominator. The denominator is \( \sqrt{3} - \sqrt{5} \). The conjugate of this expression is \( \sqrt{3} + \sqrt{5} \).
### Step 2: Multiply the Numerator and Denominator by the Conjugate of the Denominator
We multiply both the numerator and the denominator by the conjugate \( \sqrt{3} + \sqrt{5} \):
[tex]\[ \frac{2}{\sqrt{3} - \sqrt{5}} \times \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}} = \frac{2 (\sqrt{3} + \sqrt{5})}{(\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5})} \][/tex]
### Step 3: Simplify the Denominator Using the Difference of Squares Formula
The product of the denominator \( (\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5}) \) can be simplified using the difference of squares formula:
[tex]\[ (\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5}) = (\sqrt{3})^2 - (\sqrt{5})^2 \][/tex]
Evaluate the squares:
[tex]\[ (\sqrt{3})^2 = 3 \][/tex]
[tex]\[ (\sqrt{5})^2 = 5 \][/tex]
So,
[tex]\[ (\sqrt{3})^2 - (\sqrt{5})^2 = 3 - 5 = -2 \][/tex]
### Step 4: Simplify the Numerator
Next, we need to multiply the numerator by the conjugate:
[tex]\[ 2 (\sqrt{3} + \sqrt{5}) = 2\sqrt{3} + 2\sqrt{5} \][/tex]
### Step 5: Combine the Results
Now, we put the simplified numerator and the simplified denominator together:
[tex]\[ \frac{2 (\sqrt{3} + \sqrt{5})}{3 - 5} = \frac{2\sqrt{3} + 2\sqrt{5}}{-2} = \frac{7.936237570137334}{-2.000000000000001} \][/tex]
Hence, the expression simplifies to:
[tex]\[ -3.9681187850686643 \][/tex]
