Understanding the square of the group of units for 3-adic numbers

Question

My final goal is to understand the group $\mathbb{Z}_3^*/(\mathbb{Z}_3^*)^3$. However, I am having a hard time understanding how exactly an element in $(\mathbb{Z}_3^*)^3$ looks like. In this case $\mathbb{Z}_3$ is the ring of 3-adic numbers. I'm quite lost in this and any hint, help or reference for this will be helpful. Thanks!