Start by simplifying the terms under the square roots:

√96 = √(16 x 6) = 4√6

√192 = √(64 x 3) = 8√3

√54 = √(9 x 6) = 3√6

Substituting these back into the expression:

14 /108-√96+√192-√54 = 14 / 108 - 4√6 + 8√3 - 3√6

We can simplify the denominator by factoring out a common factor of 12 from each term:

108 - 4√6 + 8√3 - 3√6 = 12(9 - √6 + 2√3 - 1√6)

Now we can simplify the expression further by combining like terms:

12(9 - √6 + 2√3 - 1√6) = 12(9 + √3 - 2√6)

So the expression simplifies to:

14 / (12(9 + √3 - 2√6))

To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator, which is:

12(9 + √3 - 2√6) * (9 - √3 - 2√6)

Expanding the denominator:

12((9)^2 - (√3)^2 - (2√6)^2)

= 12(81 - 3 - 24)

= 12(54)

= 648

Multiplying the numerator and denominator by the conjugate:

14 * (9 - √3 - 2√6)

= 126 - 14√3 - 28√6

Therefore, the expression simplifies to:

(126 - 14√3 - 28√6) / 648

And we can simplify this expression by dividing the numerator and denominator by their greatest common factor (GCF), which is 14:

(126 - 14√3 - 28√6) / 648 = (9 - √3 - 2√6) / 54