20 of 100: So Many Threes

What is the last digit of: $\huge \color{#3D99F6}{3}^{\color{#D61F06}{10000}}\color{#333333}?$

Find a pattern! There are a lot of patterns like this that you can find when calculating numerical powers and products, why is that?

Bonus challenge: How many 9's appear in the final product of $\underbrace{9999 \cdots 9}_{\text{10000 9's}}\ \times 9 \,?$