The only way to simplify radicals is to take the radicand (the number under the radical) and split it into two factors, where one of them has to be a #"perfect square"#
A #"perfect square"# is a product of two of the same numbers
Example: #9# is a #"perfect square"# because #3*3=9#
So, let's simplify and pull some numbers out of these radicals:
#3sqrt(12) + 4sqrt(18)# #color(blue)(" Let's start with the left side"#
#3sqrt(4*3) + 4sqrt(18)# #color(blue)(" 4 is a perfect square")#
#3*2sqrt(3) + 4sqrt(18)# #color(blue)(" 4 is a perfect square, so take a 2 out")#
#6sqrt(3) + 4sqrt(18)# #color(blue)(" Simplify: "3*2=6," and leave the 3")#
#6sqrt(3) + 4sqrt(9*2)# #color(blue)(" 9 is a perfect square")#
#6sqrt(3) + 4*3sqrt(2)# #color(blue)(" 9 is a perfect square, so take a 3 out")#
#6sqrt(3) + 12sqrt(2)# #color(blue)(" Simplify: "4*3=12," and leave the 2")#
#color(red)(6sqrt(3) + 12sqrt(2))#
Since #sqrt(3)# and #sqrt(2)# are different radicals, we can't add them, so we're done.
