What is the digit in the unit place of $(232{)}^{33}$?

To find the unit digit of $(232{)}^{33}$, we only need to look at the unit digit of the base, which is $2$, and the exponent, which is $33$.
The unit digits of powers of $2$ follow a repeating pattern of cyclicity of $4$:

* $2^1=2$
* $2^2=4$
* $2^3=8$
* $2^4=16$ (unit digit $6$)

Divide the exponent $33$ by $4$:

$$33÷4=8\text{ with a remainder of }1$$

Since the remainder is $1$, the unit digit will be the same as the unit digit of $2^1$, which is $2$.