Example 1
f(x)= m16
f(x)= m×m = m2.
m2×m2 = m4
m4×m4 = m8
m8 ×m8 = m16
Notice that this new method exploits in its calculation, the fact that mn ×mn =m2n.. This is a well known property of the exponential function and in this paper, this property will be applied to all cases of exponential functions including functions that have exponents that do not belong to the 2n sequence.
When two exponential functions are multiplied together, and the bases are similar, then we can add the exponents together. Therefore, instead of multiplying m by itself 15 times, we can multiply the results of every multiplication by itself and we will end up with only four distinct multiplication operations. Therefore we have reduced the number of multiplication operations from 15 to 4. We can calculate the percentage of reduction in inefficiency in multiplication operations. 15-4 = 11. Therefore, we have reduced the number of multiplication operations by 11. Therefore, in percentage terms we have reduced inefficient multiplication operations by (11÷15)×100= 73.33%. This is a 73.33% reduction in inefficient multiplication operations and it is a significant improvement in efficiency. What is magical about this method is the fact that the reduction in inefficiency actually increases as the value of the exponent keeps on increasing. I will give a second example with a slightly larger exponent value to prove this salient point.