This is 1.5.8 on page 15 of Dixmier's _C* Algebra_:
Let $A$ be a C*-algebra. For each positive integer $n$, we have $A = A^n$. It is enough to show that each hermitian element $x$ of $A$ is a product of $n$ elements of $A$. Now, if $f_1, \cdots, f_n$ are continuous real-valued functions of a real variable such that $f_1(t)f_2(t) \cdots f_n(t) =t$, and $f_1(0) = \cdots = f_n(0)=0$ then $x=f_1(x) \cdots f_n(x)$.
Here, an element $x$ in a C*-algebra is hermitian if $x^* =x$.
I don't why can the author assume the existence of these functions $f_1, \cdots, f_n$. Would anyone please give me a reason or a method to construct them? Thanks a a lot.