The unit's digit of $12^{12}$ is equal to:

When finding the unit's digit of any number, only the unit's digit of the base number matters.

Since $12$ ends in $2$, we only need to find the pattern of unit's digits for powers of $2$.

---

The first few powers of $2$ and their unit's digits:

$2^1 = 2$ (unit's digit: $2$)

$2^2 = 4$ (unit's digit: $4$)

$2^3 = 8$ (unit's digit: $8$)

$2^4 = 16$ (unit's digit: $6$)

$2^5 = 32$ (unit's digit: $2$)

$2^6 = 64$ (unit's digit: $4$)

The unit's digits repeat in a cycle: $2 \to 4 \to 8 \to 6 \to 2 \to 4 \to 8 \to 6...$

This cycle has length $4$.

---

To find the unit's digit of $2^{12}$, divide the exponent by $4$:

$12 \div 4 = 3$ with remainder $0$

When the remainder is $1$: unit's digit is $2$ (1st position)

When the remainder is $2$: unit's digit is $4$ (2nd position)

When the remainder is $3$: unit's digit is $8$ (3rd position)

When the remainder is $0$: unit's digit is $6$ (4th position)

---

Since the remainder is $0$, the unit's digit corresponds to the 4th position of the cycle.

Therefore, the unit's digit of $12^{12}$ is $6$.