On the symbol length of symbols in Galois cohomology

Let $F$ be a field with absolute Galois group $G_F$, $p$ be a prime, and $\mu_{p^e}$ be the $G_F$-module of roots of unity of order dividing $p^e$ in a fixed algebraic closure of $F$.
Let $\alpha \in H^n(F,\mu_{p^e}^{\otimes n})$ be a symbol (i.e $\alpha=a_1\cup \dots \cup a_n$ where $a_i\in H^1(F, \mu_{p^e})$) with effective exponent $p^{e-1}$ (that is $p^{e-1}\alpha=0 \in H^n(G_F,\mu_p^{\otimes n})$. In this work we show how to write $\alpha$ as a sum of symbols from $H^n(F,\mu_{p^{e-1}}^{\otimes n})$. If $n>3$ and $p\neq 2$ we assume $F$ is prime to $p$ closed.