how to simplify 2000^1/3?

2000^(1/3)
= ³√(2000)
= ³√(2 * 10^3)
= 10³√2

Break it down to all its prime factors, then find all the ones that come in triplets.

2000 = 2*2*2*2*5*5*5
(I use * as a multiplication sign)

There are two triplets:

2000 = 2*(2*2*2)*(5*5*5)

We use these two rules:
1) the cube root can be distributed to the factors
example: CubeRoot(216) = CubeRoot(8*27) = CubeRoot(8) * CubeRoot(27) = 2*3 = 6

2) the cube root of a triplet of factors is simply one of the factors:
example 125 = 5*5*5
CubeRoot(125) = CubeRoot(5*5*5) = 5

Therefore
CubeRoot(2000) = CubeRoot[2*(2*2*2)*(5*5*5)]
(rearranging the order does not change the product)
= CubeRoot[(2*2*2)*(5*5*5)*2]
Next, we "distribute" the cube root:
= CubeRoot(2*2*2) * CubeRoot(5*5*5) * CubeRoot(2)
We solve the triplets:
= 2*5*CubeRoot(2)
=10*CubeRoot(2)

³√2000
= ³√(1000 * 2)
RULE: = √(a*b) = √a*√b so
= ³√(1000) * ³√(2)
= 10 * ³√(2)
= 10 * 2^(1/3)

10^2^(1/3) = 12.5992

10^2^(1/3)

10 cube root (2)

10*2^(1/3)