To find the unit digit of the expression $6^{129} \times 7^{307}$, we need to determine the unit digits of $6^{129}$ and $7^{307}$ separately and then multiply them.

Step 1: Find the unit digit of $6^{129}$ The unit digit of any positive integer power of $6$ is always $6$. This is because $6 \times 6 = 36$, $36 \times 6 = 216$, and so on. Therefore, the unit digit of $6^{129}$ is $6$.

Step 2: Find the unit digit of $7^{307}$ The unit digits of powers of $7$ follow a repeating cycle of $4$:

• $7^1 = 7$ (unit digit is $7$)
• $7^2 = 49$ (unit digit is $9$)
• $7^3 = 343$ (unit digit is $3$)
• $7^4 = 2401$ (unit digit is $1$)
• $7^5 = 16807$ (unit digit is $7$, and the cycle repeats)

To find where $7^{307}$ falls in this cycle, we divide the exponent $307$ by the cycle length, which is $4$: $307 \div 4 = 76 \text{ with a remainder of } 3$

A remainder of $3$ means the unit digit corresponds to the 3rd position in the cycle, which is the same as the unit digit of $7^3$. Therefore, the unit digit of $7^{307}$ is $3$.

Step 3: Multiply the unit digits Now, we multiply the unit digits obtained from both parts: $6 \times 3 = 18$

The unit digit of this product is $8$.

Thus, the digit in the unit place of the number $6^{129} \times 7^{307}$ is $8$.