Answer:
The statement is true only when both (1) and (2) are valid. If only one of (1) and (2) is valid, them the statement is not true.
Step-by-step explanation:
(1) alone is not sufficient, 27 is 3³ but is not a 12th power
(2) alone is not sufficient either, 81 is 3³ but it is not a 12th power
If both (1) and (2) are valid, then for each prime p that divides x, p should divide y and z, with y³ = x and z⁴=x.
Lets suppose that k is the highest power of p that divides y and m is the highest power that divides z, then (p^k)³ = (p^m)⁴. Therefore
p^3k = p^4m
This means that the power of p that appears on x is a multiple of both 3 and 4. Since those numbers are coprime, then that power is a multiple of 12.
This ensures that every prime dividing x has at least a power of 12 in the prime factirization, hence x is a 12th power.
