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

Aug 14, 2016

$2 \sqrt{x} \left(13 x - 12 y + 18\right)$

Explanation:

$\left(2 \sqrt{4 {x}^{2} + 36 x + 81} + 3 \sqrt{9 {x}^{2} - 24 x y + 16 {y}^{2}}\right) \left(2 \sqrt{x}\right)$

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.

=$\left(2 {\sqrt{\left(2 x + 9\right)}}^{2} + 3 {\sqrt{\left(3 x - 4 y\right)}}^{2}\right) \left(2 \sqrt{x}\right)$

=$\left(2 \left(2 x + 9\right) + 3 \left(3 x - 4 y\right)\right) 2 \sqrt{x}$

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

=$\left(13 x - 12 y + 18\right) 2 \sqrt{x}$

=$2 \sqrt{x} \left(13 x - 12 y + 18\right)$