By Mike O’Hern, Center Director of Mathnasium of West Knoxville
If you’ve been reading my articles for any length of time, you’ll know that I’m not big on memorizing things, but would much rather understand them. I don’t deny that to have “number facts” close at hand is a good thing for example. Yet, rather than simply memorizing them, a student should understand how to arrive at them. When the inevitable lapse of memory rears it’s head, we’re not stuck and can quickly arrive at the answer.
When I was working with a student recently, I needed to stop him for a moment to help him understand something. He was dealing with an exponent and a very special one in particular. What’s an exponent, you ask? It’s the symbol or number we put in superscript after a number or variable to tell to what power that number or variable should be raised. Clear? No? Fear not, here we go…
In the number 43, the 3 is the exponent, and it means we want 4 to the third power which means we multiply with fours three times like so: 4 x 4 x 4 = 4³ = 64. Try another? 2^8 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256.
Okay, that’s pretty clear, but my student was dealing with an exponent of zero such as 5^0. What the heck? I guess that would just be zero, like 5^0 = 0, right? That seems intuitively correct since we’re multiplying 5 zero times, but it turns out that 5^0 = 1! WHAT THE HECK??
This is where it’s important to understand rather than just memorize. When you just memorize that anything to the zero power equals 1, you’ll later come up against it and say, “What was that rule again?” After that you’ll have to research it again. But if we can arrive at the solution, we can figure it out each time we forget the answer.
…it’s always better to understand how to arrive at an answer than to memorize a rule!
Here’s how it works. We’ll do it with tens because tens are easy, but it works with any number at all. So 103 = 10 x 10 x 10 = 1,000. But 10^2 = 10 x 10 = 100, and 1,000 ÷ 10 = 100. So we see that when we divide 10^3 by 10 we get 10^2, right? As it turns out, when we divide any number raised to a power by the number itself, we simply subtract 1 from the exponent! 10^3 ÷ 10 = 10^3-1 = 10^2. Try it with 7: 7^3 = 343. 7^2 = 49. 7^3 ÷ 7 = 49 = 7^2.
So go back to the tens and see that 10^3 ÷ 10 = 10^2 = 100, so 10^2 ÷ 10 = 10^1 = 10. Each time we simply subtract 1 from the exponent. So here’s the money shot: 10^1 ÷ 10 = 1 = 10^0. Boom. 7^1 ÷ 7 = 1 = 7^0. So 4,523,906^0 = 1? Now you’ve got it.
But guess what! This can also take us into negative exponents! But we shall save that for another day. For the moment I hope you’ve learned something new. Two things, actually. First, any number raised to the zero power equals 1. Second, and more important, is that it’s always better to understand how to arrive at an answer than to memorize a rule!
Mike O’Hern, Center Director of Mathnasium of West Knoxville, earned his Bachelor’s Degree in Metallurgical Engineering at the University of Tennessee, Knoxville in 1988. He pursued graduate studies in Materials Science & Engineering while on the Research Staff at Oak Ridge National Laboratory. Mike has had a life-long love of mathematics and teaching, and feels that math is not about learning to be ready for the next math class – it’s about learning to think.