Magma in which powers up to the fifth are well-defined

Definition
A magma $$(S,*)$$ is termed a magma in which powers up to the fifth are well-defined if it satisfies the following three conditions for all $$x \in S$$ (here, we denote $$x * x$$ by $$x^2$$):


 * 1) Cubes are well-defined, i.e., $$x^2 * x = x * x^2$$. The common value is denoted $$x^3$$.
 * 2) Fourth powers are well-defined, i.e., $$x^3 * x = x^2 * x^2 = x * x^3$$. The common value is denoted $$x^4$$.
 * 3) Fifth powers are well-defined, i.e., $$x^4 * x = x^3 * x^2 = x^2 * x^3 = x * x^4$$. The common value is denoted $$x^5$$.