Asked by knmk
What is the following product?
RootIndex 3 StartRoot 5 EndRoot times StartRoot 2 EndRoot
RootIndex 6 StartRoot 10 EndRoot
RootIndex 6 StartRoot 200 EndRoot
RootIndex 6 StartRoot 500 EndRoot
RootIndex 6 StartRoot 100000 EndRoot
All Answers 1
Answered by
GPT-5 mini
AI
Rewrite everything with exponent 1/6 (and 1/3 = 2/6, 1/2 = 3/6):
cube root(5) = 5^(2/6), sqrt(2) = 2^(3/6), and the four sixth roots combine to sixth root(10·200·500·100000) = (10·200·500·100000)^(1/6) = (10^11)^(1/6).
So the product is
5^(2/6)·2^(3/6)·10^(11/6) = (5^2·2^3·10^11)^(1/6)
= (2·10^13)^(1/6).
Equivalently,
= 100·20^(1/6).
Thus the product = 100·sixth-root(20).
cube root(5) = 5^(2/6), sqrt(2) = 2^(3/6), and the four sixth roots combine to sixth root(10·200·500·100000) = (10·200·500·100000)^(1/6) = (10^11)^(1/6).
So the product is
5^(2/6)·2^(3/6)·10^(11/6) = (5^2·2^3·10^11)^(1/6)
= (2·10^13)^(1/6).
Equivalently,
= 100·20^(1/6).
Thus the product = 100·sixth-root(20).
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