Answer:
3125*k^9 + y^3 is an integer my closure property.
but 5^(1/3) is not an integer, which forces z to be irrational.
Note that there is no way an integer value can rationalize 5^(1/3)
Step-by-step explanation:
x^3 = 25z^3 - 5y^3
x^3 = 5 ( 5z^3 - y^3)
x = (5 ( 5z^3 - y^3) )^(1/3) must be an integer
= 5^(1/3) * (5z^3 - y^3)^(1/3)
Then (5z^3 - y^3)^(1/3) = 25*k^3 for some integer k
5z^3 - y^3 = 15625*k^9
5z^3 = 15625*k^9 + y^3
z^3 = 3125*k^9 + (1/5)*y^3
z = ( 3125*k^9 + (1/5)*y^3 )^ ( 1/3)
