Question

What is the following product?

RootIndex 3 StartRoot 16 x Superscript 7 Baseline EndRoot times RootIndex 3 StartRoot 12 x Superscript 9 Baseline EndRoot
x squared (RootIndex 3 StartRoot 28 x squared EndRoot)
x Superscript 5 Baseline (RootIndex 3 StartRoot 28 x EndRoot)
4 x squared (RootIndex 3 StartRoot 3 x squared EndRoot)
4 x Superscript 5 Baseline (RootIndex 3 StartRoot 3 x EndRoot)

Answer 1

First rewrite the product as
cube_root(16 x^7) * cube_root(12 x^9) * x^2·cube_root(28 x^2) * x^5·cube_root(28 x)
* 4x^2·cube_root(3 x^2) * 4x^5·cube_root(3 x).

Collect the ordinary factors: 4·4 = 16 and x^2·x^5·x^2·x^5 = x^14, so outside factor = 16 x^14.

Combine the cube roots into one:
cube_root(16·12·28·28·3·3 · x^(7+9+2+1+2+1)) = cube_root(1,354,752 · x^22).

Factor the numeric part: 1,354,752 = 2^10·3^3·7^2.
Extract perfect cubes: 2^10 = (2^9)·2, 3^3 is a cube, 7^2 remains. Also x^22 = (x^21)·x.
So cube_root(1,354,752 x^22) = (2^3·3·x^7) · cube_root(2·7^2·x) = 24 x^7 · cube_root(98 x).

Multiply by the outside 16 x^14:
16 x^14 · 24 x^7 · cube_root(98 x) = 384 x^21 · cube_root(98 x).

Answer: 384 x^21 · cube_root(98 x).