Alright, let's solve this problem step by step. We have 577 and 5021 and we're adding them together.

Step 1: We'll start by adding the digits 7 & 1 in column 1 and get 8.

Step 2: We'll start by adding the digits 7 & 2 in column 2 and get 9.

Step 3: We'll start by adding the digits 5 & 0 in column 3 and get 5.

Step 4: We'll start by adding the digits 0 & 5 in column 4 and get 5.

So, 577 + 5021 = 5598.

