Question

What is the following sum? Assume x greater-than-or-equal-to 0 and y greater-than-or-equal-to 0

StartRoot x squared y cubed EndRoot + 2 StartRoot x cubed y Superscript 4 Baseline EndRoot + x y StartRoot y EndRoot
x squared y squared StartRoot y EndRoot minus 2 x y squared StartRoot x EndRoot
2 x y StartRoot y EndRoot + 2 x y squared StartRoot x EndRoot
4 x y StartRoot x Superscript 5 Baseline y Superscript 8 Baseline EndRoot
2 x y StartRoot x y EndRoot

Answer 1

First use x,y ≥ 0 so you can pull squares out of square roots: sqrt(x^2)=x, sqrt(y^4)=y^2, etc.

Simplify each radical:
- sqrt(x^2 y^3) = x y sqrt(y)
- 2 sqrt(x^3 y^4) = 2·x y^2 sqrt(x)
- x y sqrt(y) remains
- x^2 y^2 sqrt(y) remains
- -2 x y^2 sqrt(x) remains
- 2 x y sqrt(y) remains
- +2 x y^2 sqrt(x) remains
- 4 x y sqrt(x^5 y^8) = 4 x y ·(x^2 y^4 sqrt(x)) = 4 x^3 y^5 sqrt(x)
- 2 x y sqrt(x y) = 2 x y sqrt(x) sqrt(y)

Cancel the -2 x y^2 sqrt(x) with +2 x y^2 sqrt(x). Combine the three x y sqrt(y) terms (1 + 1 + 2 = 4):

So the sum simplifies to
4 x y sqrt(y) + x^2 y^2 sqrt(y) + 2 x y^2 sqrt(x) + 4 x^3 y^5 sqrt(x) + 2 x y sqrt(x y).

You can also factor x y if you like:
x y [ (4 + x y) sqrt(y) + 2 y(1 + 2 x^2 y^3) sqrt(x) + 2 sqrt(x y) ].