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We are given the expression:

\[
\frac{2\sqrt{3} + a}{\sqrt{3} - 1}
\]

where \(a\) and \(b\) are integers, and we are tasked with simplifying the expression.

### Step 1: Rationalize the denominator
To simplify this expression, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{3} - 1\) is \(\sqrt{3} + 1\).

We multiply as follows:

\[
\frac{2\sqrt{3} + a}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1}
\]

### Step 2: Simplify the denominator
The denominator becomes a difference of squares:

\[
(\sqrt{3} - 1)(\sqrt{3} + 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2
\]

### Step 3: Expand the numerator
Now, multiply the numerator:

\[
(2\sqrt{3} + a)(\sqrt{3} + 1) = 2\sqrt{3} \cdot \sqrt{3} + 2\sqrt{3} \cdot 1 + a \cdot \sqrt{3} + a \cdot 1
\]

This simplifies to:

\[
= 2(3) + 2\sqrt{3} + a\sqrt{3} + a = 6 + (2 + a)\sqrt{3} + a
\]

### Step 4: Combine the terms
The expression becomes:

\[
\frac{6 + (2 + a)\sqrt{3} + a}{2}
\]

### Step 5: Split the expression
We can now split the expression as follows:

\[
= \frac{6}{2} + \frac{(2 + a)\sqrt{3}}{2} + \frac{a}{2}
\]

This simplifies to:

\[
= 3 + \frac{(2 + a)\sqrt{3}}{2} + \frac{a}{2}
\]

Since \(a\) is an integer, \(\frac{a}{2}\) will only be simplified if \(a\) is an even number.

Thus, the simplified expression is:

\[
3 + \frac{(2 + a)\sqrt{3}}{2} + \frac{a}{2}
\]

---

Let me know if you'd like further details or have any questions.

Here are 5 related questions:
1. How do you simplify expressions with irrational denominators in general?
2. Can you simplify the expression if \(a = 2\)? What about \(a = 4\)?
3. What is the significance of using the conjugate when simplifying fractions with square roots?
4. How would this expression change if the denominator were \(\sqrt{3} + 1\)?
5. Can you apply the same method for complex numbers in the denominator?

**Tip:** Multiplying by the conjugate is a useful trick to eliminate square roots in the denominator, making expressions easier to simplify.

Given that a, b are integers, simplify the following: (2√3 + a) / (√3 - 1).

This page explains how to simplify the algebraic expression (2√3 + a) / (√3 - 1) by rationalizing the denominator. Learn the step-by-step process of multiplying by the conjugate and applying the difference of squares formula. Suitable for students in Grades 9-11.

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