Extracting the most obvious cube:

#color(white)("XXX")216,000 = 216 xx 10^3#

Then factor out basic primes until we get to a complete factoring:

#color(white)("XXX")=2xx108xx10^3#

#color(white)("XXX")=2^2xx54xx10^3#

#color(white)("XXX")=2^3xx27xx10^3#

#color(white)("XXX")=2^3xx3xx9xx10^3#

#color(white)("XXX")=2^3xx3^3xx10^3#

#color(white)("XXXXXX")#In practice we probably would have recognized the cube factors before going this far.

Therefore

#color(white)("XXX")root(3)(216,000) = root(3)(2^3xx3^3xx10^3) = 2xx3xx10 =60#