Recently I was driving back from some hot springs with my finacee. She had fallen asleep so I was trying to pass the time by doing math stuff. I was trying to calculate in my head the square root of random numbers that I saw on the highway that were not perfect roots. Obviously I was not very successful because that is hard to do in your head.
But while I was playing these games I discovered a weird trick for doing exponents. Somebody has probably already discovered this, but after a 30 second search on Google I could not find a mention of it. So you might know this and will think I’m ignorant. But since its not easy to find on Google, hopefully this will put it at access to other people.
Start with 0^2 and 1^2.
0^2=0 and 1^2=1. The difference between those two numbers is 1. Now go to next set. 2^2=4 and 1^2=1. The difference is 3. 3^2=9 and 2^2=4. The difference is 5.
It keeps going, incrementing by odd numbers. Pretty cool right?
Check this out.
In the case of our first pair (0^2=0 and 1^2=1) add together the base number. That gives us one, which is the difference between the two solutions.
If we take 2^2=4 and 1^2=1 an addition of their base numbers gives us three, the next increment. And that keeps going on forever. But what is even cooler is that three is the number that you need add to 1^2 in order to get the answer to 2^2.
This makes it really easy to do exponents if you know the exponent right before. For example: 101^2. I don’t want to do that off the top of my head. I know that 100^2=10,000. 100+101 equals 201. Add that to 10,000 and we get 10,201, the solution to 101^2. You can do that with any pair of sequential numbers.
41^2? 40^2=1600. Add together 40 and 41. You get 81. Add that to 1600. The solution to 41^2 is 1681.