Question

What is the following sum?

3 b squared (RootIndex 3 StartRoot 54 a EndRoot) + 3 (RootIndex 3 StartRoot 2 a b Superscript 6 Baseline EndRoot)
6 b squared (RootIndex 3 StartRoot 2 a EndRoot)
12 b squared (RootIndex 3 StartRoot 2 a EndRoot
6 b squared (RootIndex 6 StartRoot 2 a EndRoot)
12 b squared (RootIndex 6 StartRoot 2 a EndRoot)

Answer 1

Interpreting the terms as

3 b^2 ∛(54a) + 3 ∛(2ab^6) + 6 b^2 ∛(2a) + 12 b^2 ∛(2a) + 6 b^2 ⁶√(2a) + 12 b^2 ⁶√(2a),

use ∛(54a)=∛(27·2a)=3∛(2a) and ∛(2ab^6)=b^2∛(2a). Then the cube-root terms become
9b^2∛(2a)+3b^2∛(2a)+6b^2∛(2a)+12b^2∛(2a)=30b^2∛(2a).

The sixth-root terms combine to 6b^2·⁶√(2a)+12b^2·⁶√(2a)=18b^2·⁶√(2a).

So the sum is
30 b^2 ∛(2a) + 18 b^2 ⁶√(2a)
= 6 b^2 (5 ∛(2a) + 3 ⁶√(2a)).