Question

Fill in the values in the algorithm of fast exponentiation:

Answer 1

The algorithm for fast exponentiation involves breaking down the exponent into its binary representation and using that to compute the result. It uses squaring and multiplication to efficiently calculate large exponentiations.

Step-by-step explanation:

The algorithm for fast exponentiation involves breaking down the exponent into its binary representation and using that to compute the result. Here is the step-by-step algorithm:

1. Convert the exponent to binary.
2. Start with the base number and square it.
3. If the first bit of the binary exponent is 1, multiply the result by the base.
4. Keep squaring the result and multiplying by the base for each subsequent bit set to 1 in the binary exponent.
5. Continue until all bits of the binary exponent have been processed.

For example, to calculate 5^9 using fast exponentiation:

1. Converting 9 to binary gives us 1001.
2. Start with the base number 5.
3. Square it to get 25.
4. Multiply the result by the base because the first bit is 1: 25 * 5 = 125.
5. Square the result to get 15625.
6. Continue to square and multiply until all bits have been processed: 15625 * 15625 = 244140625.