<Question>
The operation $\&$ is defined for positive integers $a$ and $b$ as $a \& b = \displaystyle\frac{\sqrt{a b + a}}{\sqrt{a b - b}}$. What is the value of $9 \& 2$? Express your answer as a common fraction in simplest radical form.
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find the value of 9 & 2 using the operation defined as a & b = (√(ab + a)) / (√(ab - b)). Hmm, first, I should make sure I understand the operation correctly. So, for any positive integers a and b, the operation & takes them and computes the square root of (ab + a) divided by the square root of (ab - b). Alright, so it's a fraction where both the numerator and the denominator are square roots involving products of a and b, plus or minus b.

Let me write down the expression for 9 & 2 specifically. That would be:

9 & 2 = (√(9*2 + 9)) / (√(9*2 - 2))

Let me compute the numerator and denominator step by step.

<prompt_1.1> Calculate the numerator and denominator of the expression for 9 & 2. </prompt_1.1>
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Alternatively, maybe there's a way to simplify the expression before plugging in the numbers? Let me see. Let's try to simplify the general expression for a & b first.

<prompt_1.2> Simplify the general expression for the operation & before substituting a = 9 and b = 2. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
First, calculate the numerator inside the square root: ab + a. Here, a is 9 and b is 2. So, 9*2 is 18, and then adding 9 gives 18 + 9 = 27. So the numerator becomes √27.

Now the denominator inside the square root: ab - b. Again, a is 9, b is 2. So 9*2 is 18, and subtracting 2 gives 18 - 2 = 16. So the denominator becomes √16.

Therefore, 9 & 2 simplifies to √27 divided by √16.
</execution_1.1>

<execution_1.2>
Given a & b = √(ab + a) / √(ab - b). Let's factor out common terms in the numerator and denominator under the square roots.

In the numerator: ab + a = a(b + 1). So, √(a(b + 1)).

In the denominator: ab - b = b(a - 1). So, √(b(a - 1)).

Therefore, the expression can be rewritten as:

√[a(b + 1)] / √[b(a - 1)] = √[ (a(b + 1)) / (b(a - 1)) ) ] because √x / √y = √(x/y).

So, simplifying inside the square root:

√[ (a(b + 1)) / (b(a - 1)) ) ] = √[ (a/b) * ( (b + 1)/(a - 1) ) ) ] = √(a/b) * √( (b + 1)/(a - 1) )

Hmm, maybe this can be further simplified. Let's see:

√(a/b) is the same as √a / √b. So, the entire expression becomes:

(√a / √b) * √( (b + 1)/(a - 1) ) = √a / √b * √(b + 1) / √(a - 1) = [ √a * √(b + 1) ] / [ √b * √(a - 1) ) ] = √[ a(b + 1) ] / √[ b(a - 1) ]
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Let me compute those square roots. 

<prompt_2.1> Simplify the square roots in the expression and find the final result for 9 & 2. </prompt_2.1>
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Based on execution_1.2:
But maybe plugging in a = 9 and b = 2 here would be easier.

<prompt_2.2> Substitute a = 9 and b = 2 into the simplified expression and verify the result. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
√27 is equal to √(9*3) which simplifies to 3√3. And √16 is straightforward; that's 4. So now, the expression becomes 3√3 divided by 4, which is (3√3)/4. 

Wait, let me double-check that. The numerator: 9*2 + 9 is 18 + 9 = 27, yes. √27 is 3√3. Denominator: 9*2 - 2 is 18 - 2 = 16, √16 is 4.
</execution_2.1>

<execution_2.2>
Let's try that.

First, compute √a * √(b + 1). √9 is 3, and √(2 + 1) is √3. So that's 3 * √3. Then the denominator is √b * √(a - 1). √2 is √2, and √(9 - 1) is √8. √8 can be simplified to 2√2. So the denominator becomes √2 * 2√2 = 2 * (√2 * √2) = 2 * 2 = 4. Wait, hold on, that seems different from before. Let me check that.

Wait, if the denominator is √b * √(a - 1), which is √2 * √8. √8 is 2√2, so √2 * 2√2 = 2 * (√2 * √2) = 2 * 2 = 4. That's correct. Then the numerator is √a * √(b + 1) = √9 * √3 = 3 * √3. So the entire expression is 3√3 / 4. Which matches the previous result. So that's good.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Expressed as a common fraction, that's 3 over 4 times √3.
But the problem says to express the answer as a common fraction in simplest radical form. So, simplest radical form would likely be (3√3)/4. Let me confirm if that's the most simplified form.

√3 is already a prime radical, so we can't simplify it further. And 3 and 4 are coprime, so the fraction is in simplest terms. Therefore, the value of 9 & 2 is 3√3 divided by 4.
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Wait, let me just verify once more by plugging the original numbers into the expression. Let's redo the original calculation:

<prompt_3.1> Re-verify the original calculation for 9 & 2 and confirm the final answer. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
9 & 2 = √(9*2 + 9)/√(9*2 - 2) = √(18 + 9)/√(18 - 2) = √27/√16 = 3√3 / 4. Yep, that's correct. 

I don't see any mistakes here. The key steps were computing the products and sums inside the square roots, simplifying the square roots where possible, and ensuring that the arithmetic was done correctly.
</execution_3.1>
</Execution_3>

<Final_answer>
Therefore, the value of $9 \& 2$ is $\boxed{\frac{3\sqrt{3}}{4}}$.
</Final_answer>
</think>