Solution:

Given expression:

2√2 - √2 + √3 - 6√2 / √6 - √3 - 8√3 / 6√9^2

Simplifying each part of the expression:

1. 2√2 - √2:
- We can combine like terms, which in this case are the terms with the same radicals.
- √2 is a common radical term, so we can subtract them.
- 2√2 - √2 = √2

2. √3 - 6√2 / √6:
- Here, we have a subtraction of two radicals and a division of two radicals.
- To simplify, we need to rationalize the denominator.
- Rationalizing the denominator means multiplying the numerator and denominator by the conjugate of the denominator.
- The conjugate of √6 is -√6, so we multiply the numerator and denominator by -√6.
- (√3 - 6√2) * (-√6) / (√6) * (-√6)
- (-√3√6 + 6√2√6) / -6
- (-√18 + 6√12) / -6
- Now, we can simplify the radicals and divide by -6.
- (-3√2 + 6√3) / -6
- Simplifying further, we get: -√2 + √3 / 2

3. -8√3 / 6√9^2:
- Here, we have a division of two radicals.
- We can simplify by dividing the numbers outside the radical and the numbers inside the radical separately.
- -8/6 = -4/3
- √3 / √(9^2)
- √3 / √(81)
- √3 / 9
- Simplifying further, we get: √3 / 9

Now, substituting the simplified values back into the original expression:

√2 + √3 / 2 - √2 + √3 / 9

Combining like terms:

(√2 - √2) + (√3 / 2 + √3 / 9)

Simplifying further:

0 + (9√3 + 2√3) / 18

(11√3) / 18

Therefore, the value of the given expression is (11√3) / 18.