Solve 4√6 - 19√6/-5√3

To simplify the given expression, we'll begin by simplifying each square root individually, before performing any operations.

4√6 = 4 * √(2*3) = 4 * √2 * √3 = 4√2√3 = 4√6
-19√6 = -19 * √(2*3) = -19 * √2 * √3 = -19√2√3 = -19√6

Next, we'll simplify the denominator: -5√3 = -5 * √3.

Now let's substitute these simplifications into the given expression:

(4√6 - 19√6) / (-5√3)

Combining like terms within the numerator:
= (4√6 - 19√6) / (-5√3)
= (-15√6) / (-5√3)

The -5 in the numerator and the denominator cancel out, leaving us with:
= -15√6 / √3

To rationalize the denominator, we'll multiply both the numerator and denominator by √3:
= (-15√6 / √3) * (√3 / √3)
= (-15√6 * √3) / (√3 * √3)
= (-15√18) / 3
= (-15 * √(9*2)) / 3
= (-15 * √9 * √2) / 3
= (-15 * 3 * √2) / 3
= -45√2 / 3

Finally, simplifying the fraction by dividing both the numerator and denominator by 3:
= -15√2