How would you approach multiplying two extremely large integers that exceed the normal integer ranges in an efficient manner?

To multiply two extremely large integers efficiently, you can utilize the Karatsuba algorithm.

The Karatsuba algorithm is a fast multiplication algorithm that reduces the number of multiplication operations required for multiplying two numbers by breaking them into smaller parts. Here's a step-by-step guide on how to approach multiplying two extremely large integers using the Karatsuba algorithm:

1. Break the two large integers into smaller parts by splitting them in half.
2. Recursively multiply the smaller parts until you get down to multiplication of single-digit numbers.
3. Combine the results of the smaller multiplications using addition and shifting to get the final result.

By using the Karatsuba algorithm, you can significantly reduce the number of multiplication operations needed to multiply two extremely large integers, making the process more efficient and faster.

Implementing this algorithm in your code will ensure that you can handle extremely large integers that exceed the normal integer ranges efficiently.