To find the value of expression involving the roots of a quadratic equation, we first need to find the roots of the given quadratic equation. Let's consider the quadratic equation given:

x^2 sqrt6x 3 = 0



We can solve the given quadratic equation by factoring, completing the square, or using the quadratic formula. However, the given equation seems to be incomplete, as there is no coefficient of x^2 term. Therefore, let's assume the coefficient of x^2 term is 1:

x^2 + sqrt(6)x + 3 = 0

Now, we can proceed to find the roots of the quadratic equation. By factoring, completing the square, or using the quadratic formula, we get:

x = (-sqrt(6) ± sqrt(6 - 4(1)(3))) / (2(1))
= (-sqrt(6) ± sqrt(-12)) / 2
= (-sqrt(6) ± 2i√(3)) / 2
= -sqrt(6)/2 ± i√(3)

Therefore, the roots of the quadratic equation are:

a = -sqrt(6)/2 + i√(3)
b = -sqrt(6)/2 - i√(3)



Now, let's evaluate the given expression:

a^23 b^23 a^14 b^14 / a^15 b^15 a^10 b^10

We can simplify this expression by canceling out common factors:

a^23 b^23 a^14 b^14 / a^15 b^15 a^10 b^10
= (a^(23+14) b^(23+14)) / (a^(15+10) b^(15+10))
= (a^37 b^37) / (a^25 b^25)
= a^(37-25) b^(37-25)
= a^12 b^12

Therefore, the simplified expression is a^12 b^12.

In conclusion, the given quadratic equation has roots a = -sqrt(6)/2 + i√(3) and b = -sqrt(6)/2 - i√(3). The expression involving these roots simplifies to a^12 b^12.