Asked by vt5

What is the following difference?

2 a b (RootIndex 3 StartRoot 192 a b squared EndRoot) minus 5 (RootIndex 3 StartRoot 81 a Superscript 4 Baseline b Superscript 5 Baseline EndRoot)
Negative 3 a b (RootIndex 3 StartRoot 3 a b squared EndRoot)
16 a b squared (RootIndex 3 StartRoot 3 a EndRoot) minus 45 a squared b squared (RootIndex 3 StartRoot 3 b EndRoot)
Negative 7 a b (RootIndex 3 StartRoot 3 a b squared EndRoot)
8 a b (RootIndex 3 StartRoot 3 a b squared EndRoot) minus 15 a b squared (RootIndex 3 StartRoot 3 a b EndRoot)

All Answers 1

Answered by GPT-5 mini AI
Compute each cube root by pulling out perfect cubes:

∛(192 a b^2) = ∛(64·3·a·b^2) = 4·∛(3 a b^2), so
2ab·∛(192 a b^2) = 2ab·4·∛(3 a b^2) = 8ab·∛(3 a b^2).

∛(81 a^4 b^5) = ∛(27·3·a^3·a·b^3·b^2) = 3ab·∛(3 a b^2), so
5·∛(81 a^4 b^5) = 15ab·∛(3 a b^2).

Subtracting:
8ab·∛(3 a b^2) − 15ab·∛(3 a b^2) = −7ab·∛(3 a b^2).