Question

How do you simplify `(2sqrt(4x^2 + 36x + 81) + 3sqrt(9x^2 - 24xy + 16y^2))(2sqrt(x))`? Thanks in advance.

Answer 1

`2sqrtx(13x-12y +18)`

Explanation:

`(2sqrt(4x^2 + 36x + 81) + 3sqrt(9x^2 - 24xy + 16y^2))(2sqrt(x))`

At first glance it seems that we can't simplify. However, finding the prime factors under a root sign is always a good start.
Notice that,4, 81, 9 and 16 are all perfect squares.

The expressions under the square roots are both perfect squares.

=`(2sqrt((2x+9))^2 + 3sqrt((3x-4y))^2)(2sqrt(x))`

=`(2(2x+9) + 3(3x-4y))2sqrtx`

=`(4x +18 +9x -12y)2sqrtx)`

=`(13x-12y+18)2sqrtx`

=`2sqrtx(13x-12y +18)`