modular arithmetic - How to compute $3^9 pmod {10}$

The solution given was:

$3^9 = (27)^3 = 7^3 = 49(7) = 9(7) = 63 = 3$

I understand up to $\ 3^9 = (27)^3$ But after that I am lost. Can someone explain how to solve this and what is going on here?

Answer

The $27$ was replaced with a $7$ because $27\pmod{10}=7$. The same concept arose with the $49$ later, as $49\pmod{10}=9$, and likewise $63\pmod{10}=3$. This is why the values greater than $10$ are switched.