Question

The number of multiplications needed to find $ x^{32} $ when $ x $ is given, is

• A. 32
• B. 31
• C. 5
• D. 8

Hint:

When computing large powers, try using exponentiation by squaring to minimize the number of multiplications.

Answer 1

The Correct Option is C

To compute \( x^{32} \), we can use the technique of exponentiation by squaring, which reduces the number of multiplications required.
1. Direct Approach: If we were to compute \( x^{32} \) by repeated multiplication, we would multiply \( x \) by itself 31 times. This would require 31 multiplications. 
2. Exponentiation by Squaring: Instead of multiplying repeatedly, we can break down the exponentiation: \[ x^{32} = (x^2)^{16} \] Now, we square the result: \[ x^2 = x \times x \quad (\text{1 multiplication}) \] \[ (x^2)^2 = x^4 \quad (\text{1 multiplication}) \] \[ (x^4)^2 = x^8 \quad (\text{1 multiplication}) \] \[ (x^8)^2 = x^{16} \quad (\text{1 multiplication}) \] \[ (x^{16})^2 = x^{32} \quad (\text{1 multiplication}) \] Hence, we perform 5 multiplications in total. 
Conclusion:
The number of multiplications required to compute \( x^{32} \) is 5.