Natural Number has Same Prime Factors as Integer Power

Theorem
The nth power of any natural number has the same prime factors as the natural number.

Proof
Let $x$ be a natural number.

Let $n$ be a natural number.

Suppose that $x$ has a set of unique prime factors, such that $x = p_1p_2 \cdots p_k$.

When $x$ is raised to the $n$th power, each prime factor is raised to the $n$th power.

Thus, raising $x$ to the $n$th power introduces no new prime factors since each prime factor in $x$ appears $n$ times in $x^n$