Question

What is the following product? Assume x greater-than-or-equal-to 0

RootIndex 3 StartRoot x squared EndRoot times RootIndex 4 StartRoot x cubed EndRoot
x StartRoot x EndRoot
RootIndex 12 StartRoot x Superscript 5 EndRoot
x (RootIndex 12 StartRoot x Superscript 5 EndRoot)
x6

Answer 1

Interpret each radical as a fractional power (x ≥ 0 so no absolute values):

∛(x^2) = x^(2/3)
⁴√(x^3) = x^(3/4)
x√x = x · x^(1/2) = x^(3/2)
¹²√(x^5) = x^(5/12)
x·¹²√(x^5) = x · x^(5/12) = x^(17/12)
x^6 = x^6

Multiply ⇒ add exponents:
2/3 + 3/4 + 3/2 + 5/12 + 17/12 + 6
= (8+9+18+5+17+72)/12 = 129/12 = 43/4.

So the product = x^(43/4), equivalently x^10 · x^(3/4) = x^10 · ⁴√(x^3).