Step 5
f(x) = m22
f(x)= m×m = m2.
m2×m2 = m4
m4×m4 = m8
m8 ×m8 = m16
m16×m16 = m32
Since 32 is greater than 22, we stop the multiplication operation here.
In step 6, we will have to subtract the penultimate exponent (16) from the exponent that we are calculating (22).
Step 6 .
Hence, 22-16=6.
Therefore we can rewrite f(x)=m310 as:
f(x)=m256×m32×m16×m6.
Notice that 256, 32 and 16 are all penultimate exponents and they also belong to the 2n sequence. However, 6 is not a penultimate exponent, it is a remainder exponent because it is what remains after we have removed all possible 2n sequence exponents. This author stopped when the exponent of the value that does not belong to the 2n sequence reduced to a single digit value. The assumption is that m6 is easy to calculate no matter how large the value of m is. The final step which is step 7, involves solving for f(x)=m256×m32×m16×m6.
The beauty of this method is that once we have calculated the first round of multiplication for m310 in table 1, then we already obtained the values for m256, m32 and m16. This means that we do not need to calculate these values again. We can just go back to table 1 in example 4 and pick them up. Therefore m256, m32 and m16 need zero new multiplication operations. However, notice that m6 is not found in table 1 and we will have to calculate it manually. However it is expected that it will be trivial to calculate m6for mathematicians or computers regardless of the size of m. We know that m6 will take 5 multiplication operations to calculate it. From table1, we can see that m256, m32 and m16 require 9 multiplication operations to calculate. That is, table 1 requires 9 multiplication operations to create and once it is created, we can obtain values for m256, m32 and m16without further calculations. Therefore how many multiplication operations does it take to calculate m256×m32×m16×m6 if we already have the values for each operand separately? It takes 3 multiplication operations to calculate it. Therefore the total multiplication operations necessary to solve this problem is 9+5+3=17. Therefore instead of multiplying m by itself 309 times, we can just multiply it 17 times to get the answer if we use this new method. We will reduce the number of multiplication steps needed by 309-17=292. This is a reduction in inefficiency of 292÷309×100= 94.5%