This video also contains digits with powers, what do you do in those cases

Euler's theorem. Which you apparently either didn't apply or didn't mention.

It states that if $a$ is not divisible by $2$ or $5$, then the last digit of $a^4 = 1$.

Thus for example, to compute

$$333^{4323133}$$

we write $4323133 = 4*1080783 + 1$

Hence in mod 10

$$333^{4323133} = (333^4)^{1080783}333 = 333 = 3.$$

Isn't this a lot easier?????

Now if $a$ is divisible by $5$, then the last digit of $a$ is always $5$ or $0$, and it is easy to see which.

If $a$ is divisible by $2$, then use that the last digit of $2^5 = 2$.