Satish Kumar Nagarjan explains simple methods to find the square and cube of any two-digit number. For squaring a two-digit number like 37, he writes it as (3 7)2 and expands it using the formula (a + b)2 = a2 + 2ab + b2. For cubing a two-digit number like 37, he writes it as (3 7)3 and expands it using the formula (a + b)3 = a3 + 3a2b + 3ab2 + b3. He demonstrates these methods by calculating the square of 37 as 1369 and the cube of 37 as 50653.