Okay, let's try to figure out the value of 9 & 2 using the given operation. The operation is defined as a & b = sqrt(ab + a) divided by sqrt(ab - b). So, first, I need to plug in a = 9 and b = 2 into this formula. Let me write that out step by step.

First, let's compute the numerator inside the square root: ab + a. If a is 9 and b is 2, then ab is 9*2 = 18. Adding a (which is 9) to that gives 18 + 9 = 27. So the numerator becomes sqrt(27).

Then, the denominator inside the square root is ab - b. Again, ab is 18, and subtracting b (which is 2) gives 18 - 2 = 16. So the denominator is sqrt(16).

So putting that together, the operation 9 & 2 is sqrt(27)/sqrt(16). Now, let's simplify both square roots. sqrt(27) can be simplified because 27 is 9*3, and sqrt(9*3) is sqrt(9)*sqrt(3) = 3*sqrt(3). Similarly, sqrt(16) is 4 because 4*4=16. So substituting those back in, we have (3*sqrt(3))/4.

Wait, let me check that again. The numerator sqrt(27) is indeed 3*sqrt(3) because 27 is 3^3, so sqrt(3^3) = 3*sqrt(3). The denominator sqrt(16) is definitely 4. So dividing those gives (3*sqrt(3))/4, which is already in simplest radical form. There's no further simplification needed there.

But just to make sure I didn't make any mistakes in the initial steps. Let's verify each part again. For the numerator: ab + a = 9*2 + 9 = 18 + 9 = 27. Correct. Denominator: ab - b = 18 - 2 = 16. Correct. Then sqrt(27) is 3*sqrt(3), and sqrt(16) is 4. So the final answer is 3*sqrt(3)/4. That seems right.

Alternatively, maybe there's another way to approach this problem by simplifying the expression before plugging in the numbers. Let's see. The operation is sqrt(ab + a)/sqrt(ab - b). Maybe we can factor a and b out of the numerator and denominator inside the square roots.

Looking at the numerator: ab + a = a(b + 1). Similarly, the denominator: ab - b = b(a - 1). So the operation becomes sqrt(a(b + 1))/sqrt(b(a - 1)). Then, since both are square roots, we can combine them into a single square root of the ratio: sqrt[ (a(b + 1)) / (b(a - 1)) ). Let's see if that helps.

So substituting a = 9 and b = 2, we get sqrt[ (9*(2 + 1)) / (2*(9 - 1)) ) ] = sqrt[ (9*3) / (2*8) ) ] = sqrt[ 27 / 16 ). Which is the same as sqrt(27)/sqrt(16) = 3*sqrt(3)/4. So that confirms the same result. So either way, whether we compute numerator and denominator separately or factor first, we end up with the same answer. Therefore, I'm confident that the value of 9 & 2 is 3*sqrt(3)/4.

Just to recap all steps once more to ensure no miscalculations:

1. Compute ab + a: 9*2 + 9 = 18 + 9 = 27. sqrt(27) = 3*sqrt(3).
2. Compute ab - b: 9*2 - 2 = 18 - 2 = 16. sqrt(16) = 4.
3. Divide the two results: 3*sqrt(3)/4.
4. Check if the answer can be simplified further. Since 3 and 4 are coprime, and sqrt(3) is already simplified, there's no further simplification.
5. Verify using an alternative method (factoring numerator and denominator inside the square roots) leads to the same result. So the answer is consistent.

Therefore, after careful consideration and multiple methods of verification, I conclude that the value of 9 & 2 is indeed 3√3⁄4.
Final Answer
The value of $9 \& 2$ is \boxed{\dfrac{3\sqrt{3}}{4}}.
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