sqrt(6a) - 4sqrt(54a) - 4sqrt(216a)
sqrt(6a) - 4sqrt(9 * 6a) - 4sqrt(36 * 6a)
sqrt(6a) - 4(3)sqrt(6a) - 4(6)sqrt(6a)
sqrt(6a) - 12sqrt(6a) - 24sqrt(6a)
-35sqrt(6a)
That may be the answer your teacher is looking for, but. . .
.
sqrt(6a) - 4sqrt(54a) - 4sqrt(216a)
sqrt(6a) - 4sqrt(9 * 6a) - 4sqrt(36 * 6a)
In the problem above, with - 4sqrt(9 * 6a), we factored out sqrt(9) as 3. The trick here is that sqrt(9) is not just 3 but also -3 because (-3)^2 = 9.
This means instead of this:
sqrt(6a) - 4(3)sqrt(6a) - 4(6)sqrt(6a)
It would be more correct to write this:
sqrt(6a) +- 4(3)sqrt(6a) - 4(6)sqrt(6a)
The same thing happens with sqrt(36) being 6 and -6, so it would be even more correct to write this:
sqrt(6a) +- 4(3)sqrt(6a) +- 4(6)sqrt(6a)
The +- means that part can be + or -. Since we have two +-'s that means we have 4 answers:
sqrt(6a) + 12sqrt(6a) + 24sqrt(6a) = 37sqrt(6a)
sqrt(6a) + 12sqrt(6a) - 24sqrt(6a) = -11sqrt(6a)
sqrt(6a) - 12sqrt(6a) + 24sqrt(6a) = 13sqrt(6a)
sqrt(6a) - 12sqrt(6a) - 24sqrt(6a) = -35sqrt(6a)
So the real answer is that it can be any of these four solutions. All four of these solutions are true.
Answer: -35sqrt(6a) or -11sqrt(6a) or 13sqrt(6a) or 37sqrt(6a).