Solving radicals with addition and subtraction

Question

√12 - 2√27 + 3√75 - √48 -√80 +√20

Answer 1

√12 - 2√27 + 3√75 - √48 -√80 +√20

2√3  - 6√3  + 15√5  - 4√3  - 4√5  + 2√5

(2-6-4)√3  + (15-4+2)√5

-8√3 + 13√5

 

the answer is -8√3 + 13√5

Answer 2

√12 - 2√27 + 3√75 - √48 -√80 +√20

 

Let's start by breaking down the roots into pieces that hopefully that can be square rooted:

√(3*4) - 2√(9*3) + 3√(25 * 5) - √(16*3) - √(16*5) +√(4*5)

 

So we can now find the partial roots:

2√3 - (2*3)√3 + (3*5)√5 - 4√3 - 4√5 +2√5

2√3 - 6√3 + 15√5 - 4√3 - 4√5 +2√5

 

From here, it's just a matter of adding and subtracting the "coefficients":

13√5 - 8√3