How do you simplify (2sqrt(4x^2 + 36x + 81) + 3sqrt(9x^2 - 24xy + 16y^2))(2sqrt(x))(2√4x2+36x+81+3√9x2−24xy+16y2)(2√x)? Thanks in advance.

1 Answer
EZ as pi
Aug 14, 2016

2sqrtx(13x-12y +18)2√x(13x−12y+18)

Explanation:

(2sqrt(4x^2 + 36x + 81) + 3sqrt(9x^2 - 24xy + 16y^2))(2sqrt(x))(2√4x2+36x+81+3√9x2−24xy+16y2)(2√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)2+3√(3x−4y)2)(2√x)

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

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

=(13x-12y+18)2sqrtx(13x−12y+18)2√x

=2sqrtx(13x-12y +18)2√x(13x−12y+18)

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