Question

What is the following sum?

5 x (RootIndex 3 StartRoot x squared y EndRoot) + 2 (RootIndex 3 StartRoot x Superscript 5 Baseline y EndRoot)
7 x (RootIndex 6 StartRoot x squared y EndRoot)
7 x squared (RootIndex 6 StartRoot x y squared EndRoot)
7 x squared (RootIndex 3 StartRoot x y squared EndRoot)
7 x (RootIndex 3 StartRoot x squared y EndRoot)

Answer 1

First rewrite each radical with fractional exponents:

5x∛(x^2y) + 2∛(x^5y) + 7x⁶√(x^2y) + 7x^2⁶√(xy^2) + 7x^2∛(xy^2)
= 5x·x^(2/3)y^(1/3) + 2·x^(5/3)y^(1/3) + 7x·x^(1/3)y^(1/6) + 7x^2·x^(1/6)y^(1/3) + 7x^2·x^(1/3)y^(2/3).

Notice ∛(x^5y)=x·∛(x^2y), so the first three cube-root terms combine:
5x∛(x^2y) + 2∛(x^5y) + 7x∛(x^2y) = (5+2+7)x∛(x^2y) = 14x∛(x^2y).

Therefore the sum simplifies to either of these equivalent forms:

Radical form:
14x∛(x^2y) + 7x·⁶√(x^2y) + 7x^2·⁶√(xy^2) + 7x^2∛(xy^2),

or fractional-exponent form:
14 x^(5/3) y^(1/3) + 7 x^(4/3) y^(1/6) + 7 x^(13/6) y^(1/3) + 7 x^(7/3) y^(2/3).

You can also factor 7:
7[2x∛(x^2y) + x·⁶√(x^2y) + x^2·⁶√(xy^2) + x^2∛(xy^2)].