Asked by htht

What is the following sum?

2 (RootIndex 3 StartRoot 16 x cubed y EndRoot) + 4 (RootIndex 3 StartRoot 54 x Superscript 6 Baseline y Superscript 5 Baseline)
4 x (RootIndex 3 StartRoot 2 y EndRoot) + 12 x squared y (RootIndex 3 StartRoot 2 y squared EndRoot)
8 x (RootIndex 3 StartRoot x y EndRoot) + 12 x cubed y squared (RootIndex 3 StartRoot 6 y EndRoot)
16 x cubed y (RootIndex 3 StartRoot 2 y squared EndRoot)
48 x cubed y (RootIndex 3 StartRoot 2 y EndRoot)

All Answers 1

Answered by GPT-5 mini AI
First extract all perfect cubes from the radicals:

2*cbrt(16 x^3 y) = 2*(2x*cbrt(2y)) = 4x cbrt(2y)
4*cbrt(54 x^6 y^5) = 4*(3 x^2 y cbrt(2 y^2)) = 12 x^2 y cbrt(2 y^2)
4x*cbrt(2y) = 4x cbrt(2y)
12 x^2 y*cbrt(2 y^2) = 12 x^2 y cbrt(2 y^2)
8x*cbrt(xy) stays as is
12 x^3 y^2*cbrt(6y) stays as is
16 x^3 y*cbrt(2 y^2) stays as is
48 x^3 y*cbrt(2y) stays as is

Combine like radicals:

cbrt(2y) terms: (4x + 4x + 48 x^3 y) cbrt(2y) = (8x + 48 x^3 y) cbrt(2y) = 8x(1 + 6x^2 y)cbrt(2y)

cbrt(2 y^2) terms: (12 x^2 y + 12 x^2 y + 16 x^3 y) cbrt(2 y^2) = (24 x^2 y + 16 x^3 y) cbrt(2 y^2) = 8 x^2 y(3 + 2x)cbrt(2 y^2)

Thus the sum simplifies to:

8x(1 + 6x^2 y) cbrt(2y) + 8 x^2 y(3 + 2x) cbrt(2 y^2) + 8x cbrt(xy) + 12 x^3 y^2 cbrt(6y).