Question 1209535
Here's how to rationalize and simplify the expression:

1/√3 + 1 - (√3 - 1) / (2√3)

1. **Rationalize the first term:**
   Multiply the numerator and denominator by the conjugate of the denominator:
   (1 / (√3 + 1)) * ((√3 - 1) / (√3 - 1)) = (√3 - 1) / (3 - 1) = (√3 - 1) / 2

2. **Rationalize the second term:**
   Multiply the numerator and denominator by √3:
   ((√3 - 1) / (2√3)) * (√3 / √3) = (3 - √3) / 6

3. **Combine the terms:**
   (√3 - 1) / 2 - (3 - √3) / 6

4. **Find a common denominator (6):**
   (3(√3 - 1)) / 6 - (3 - √3) / 6 = (3√3 - 3 - 3 + √3) / 6 = (4√3 - 6) / 6

5. **Simplify:**
   (4√3 - 6) / 6 = (2√3 - 3) / 3

So, the simplified and rationalized form of the expression is (2√3 - 3) / 3.