$G$ be a finite simple group , then every element of $G$ can be written as a product of $n$-th powers of elements of $G$?

Question

Let $G$ be a finite simple group , let $n$ be a positive integer such that not all $n$-th powers of elements of $G$ are identity , then is it true that every element of $G$ can be written as a product of $n$-th powers of elements of $G$ ?

Answer 1

The following steps reach the destination.

1. The subgroup generated by the $n$th powers is non-trivial because there are non-trivial $n$th powers.
2. The conjugate of an $n$th power is another $n$th power.
3. The same holds for any product of $n$th powers, so the group generated by the $n$th powers is normal, and hence all of $G$