a) First of all, to combine the expressions we should simplify the surds as much as possible. I will break this simplification up for each term:
1. √(48n^3) = √( (4n)^(2) * 3n)
= 4n√(3n)
Here, the key is to ask yourself, what in the surd can be broken up into an even power?
Given that we have 48, we can say that 48 = 16*3 = 4^(2)*3.
We also have n^3 = n^(2)*n.
Together, 4^(2) and n^(2) can be written as (4n)^2, whilst 3 and n can be written as 3n.
Since the square root of a value squared is that value, we can then extract 4n from under the square root, and are left with 3n under the square root. Thus we have √(48n^3) = 4n√(3n)
2. Now we apply the same principle to the second surd:
√(9n) = √3^(2)*n
= 3√n
3. The third surd, √(3n), is a case where you cannot simplify any further.
b) Now that we have simplified each of the surds as much as possible, let us write out the simplified equation:
√(48n^3) + √(9n) - √(3n) = 4n√(3n) + 3√(n) - √(3n)
c) Since the terms 4n√(3n) and -√(3n) have a common factor of √(3n), we can combine them into (4n - 1)√(3n).
Note that there is a -1 in the bracket since -√(3n) is effectively equal to -1√(3n).
d) Now that we have combined those two terms, we may write out the combined expression in full:
4n√(3n) + 3√(n) - √(3n) = (4n - 1)√(3n) + 3√(n)
Thus, the second answer is correct.