Question

What is the following product?

RootIndex 3 StartRoot 4 EndRoot times StartRoot 3 EndRoot
2 (RootIndex 6 StartRoot 9 EndRoot)
RootIndex 6 StartRoot 12 EndRoot
RootIndex 6 StartRoot 432 EndRoot
2 (RootIndex 6 StartRoot 3,888 EndRoot)

Answer 1

Interpreting the expression as

∛4 · √3 · 2 · ⁶√9 · ⁶√12 · ⁶√432 · 2 · ⁶√3888,

group the sixth roots and the numeric factors:

= 4^{1/3} · 3^{1/2} · 4 · (9·12·432·3888)^{1/6}.

Factor the product inside the sixth root:
9·12·432·3888 = 2^{10}·3^{11},

so (9·12·432·3888)^{1/6} = 2^{10/6}·3^{11/6} = 2^{5/3}·3^{11/6}.

Now combine all powers:
2-exponent: 2^{2/3} (from 4^{1/3}) · 2^{5/3} · 2^2 = 2^{13/3},
3-exponent: 3^{1/2} · 3^{11/6} = 3^{7/3}.

Thus the product is 2^{13/3}·3^{7/3} = (2^{13}·3^7)^{1/3}.

Extract cubes: = 2^4·3^2·∛(2·3) = 16·9·∛6 = 144·∛6.

Answer: 144 · ∛6.