Okay, we are tasked with adding 7766 and 1990. Let's begin!

Step 1: We'll start by adding the digits 6 & 0 in column 1 and get 6.

Step 2: We'll start by adding the digits 6 & 9 in column 2 and get 15.
We'll write down the last digit 5 and carry the 1 to the next column.

Step 3: We'll start by adding the digits 7 & 9 in column 3 and get 17.
We'll write down the last digit 7 and carry the 1 to the next column.

Step 4: We'll start by adding the digits 7 & 1 in column 4 and get 9.

So, 7766 + 1990 = 9756.

