You have three factors in this product and you are only interested in the units digit, so let's start by analyzing the units digit of each of the three factors:
a. 13^4 - since you are only interested in the units digit of this one, then you are looking for the units digit of 3^4. Since 3^2 = 9 and 3^4 = (3^2)^2, then the units digit of 3^4 and 13^4 will be the units digit of 9^2 = 1.
b. 17^2 - again, you are looking for the units digit of 7^2, which will be 9 (since 7^2 = 49)
c. 29^3 - units digit of 9^3 will be 9, since the units digit of 9^2 is 1 (9^2 = 81). Multiply 1 with 9 and you get 9.

So you have the following units digits: 1, 9 and 9. Multiply these babies and the number will have a final units digit of 1 (since, again, 9*9 = 81).

Yes, there is a pattern when it comes to multiplying numbers by itself.

2,3,7 and 8 - repeats its unit digit every four times. In other words, 2^1 will have the same units digit as 2^5. 3^2 will have the same as 3^6.

A quick way to determine the units digit for a number like 7^67 is to take 67 and divide it by 4. The remainder, 3, will be have the same units digit as 7^3. Another example, 2^38 will have the same units digit as 2^2.

0,1,5 and 6 will have the same units digit.

4 and 9 alternates between two numbers. 4 will be either 4 or 6. 9 will be either 9 or 1.