To find the unit digit of $9^{1234}$, we can look for a pattern in the units digit of powers of 9.

Let’s compute the first few powers of 9 and observe the units digits:

$9^1 = 9 \quad \text{(units digit is 9)}$ $9^2 = 81 \quad \text{(units digit is 1)}$ $9^3 = 729 \quad \text{(units digit is 9)}$ $9^4 = 6561 \quad \text{(units digit is 1)}$

The units digits repeat in a cycle of 2: $9, 1$.

Now, we divide the exponent $1234$ by the cycle length (which is 2): $1234 \div 2 = 617 \text{ remainder } 0$ Since the remainder is 0, the units digit of $9^{1234}$ corresponds to the units digit of $9^2$, which is 1.

Thus, the unit digit of $9^{1234}$ is 1.

Do you want further details or have any questions?

Here are 5 related questions for further practice:

1. What is the unit digit of $7^{345}$?
2. Find the unit digit of $2^{100}$.
3. What is the unit digit of $5^{1000}$?
4. What is the unit digit of $6^{789}$?
5. Find the unit digit of $3^{4567}$.

Tip: Recognizing patterns in the unit digits of powers can significantly speed up solving large exponent problems!