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© Copyright 1999, Jim Loy
Ladies and gentlemen, or girls and gentleboys, I am about to do an amazing feat of calculation (using my highly tuned brain) right before your very eyes and ears. Now, how many of you know what a fifth power is? OK, for those of you who don't, you take a number, let's say it is 2. And you multiply it times itself five times, 2x2x2x2x2. That is 2 to the fifth power, also known as 2 to the fifth. Now, does anyone have a calculator which shows ten or more digits? OK, Susan is our official calculator operator. What is 2 to the fifth? 32. That is right. My highly tuned brain already knew that. [I write down 2 and 32, on the chalk board.] How about 3 to the fifth? 243 [write down 3 and 243]. We might as well write down a few more, just for the practice. What is 4 to the fifth? [We go up to 9 to the fifth, and we now have this table on the board:]
2 32 3 243 4 1024 5 3125 6 7776 7 16807 8 32768 9 59049
You see, my highly tuned brain already knew all of that. Pretty good, huh?
Now William, I want you to come up to the chalk board, and write down any two digit number. I will turn my back so I don't see it. OK, my back is turned. Now after everyone has seen the two digit number, erase it so I won't be able to read it. Are you done with that? Good. Now Susan, you've got the calculator. Take William's number and multiply it times itself five times. Got that? OK, now Susan, tell me your answer. [I write her answer: 656356768.] Hm, kind of big. [As I say that, I write 58, which is William's original number. The audience is amazed.]
How did I do that? Well, I knew that the last digit of the answer was 8 because the last digit of 656356768 was 8. That is how it works with fifth powers, the last digit is the same. Then I mentally crossed out the rightmost five digits, and looked at what was left: 6563. Then without making it obvious, I looked at the above table. 6563 is between 3125 and 7776, and so 5 is the left digit of the answer. The answer is 58. That is all there is to it. It would have been more amazing if I had memorized the above table. But, I keep forgetting it.
If no one had a calculator that could handle ten digits, then someone in the audience would have had to do the last step by hand: 58x58x58x58 (which is eight or fewer digits) on the calculator, and then x58 by hand. I had Susan do the simpler calculations, just to make sure that she got the routine down, and would do the same with William's two digit number.
You can do something similar with cubes and cube roots. This takes a little more memorization. In this case the table is:
2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729
If the rightmost digit of the cube is 0, 1, 4, 5, 6, or 9, then the rightmost digit of the answer is the same. If the rightmost digit of the cube is 2, 3, 7, or 8, then the right-most digit of the answer is 8, 7, 3, or 2 (in that order). We mentally cross out the rightmost three digits, and compare our remaining digits with the table, to find the leftmost digit of the answer.