Why does 2^0 equal one, and not zero (using division)?

Pre-Algebra Tutorial

Why does 2^0 equal one, and not zero (using division)?

Intro

I’m writing this tutorial to explain to younger learners why any nonzero number raised to the power of zero equals one, and not zero. That is, this question begins to address why the following equation is correct: X^0 = 1 ,for all X ≠ 0.

Specifically, I am writing this in such a way that younger students can grasp why this is the case. I do not believe in just giving students rules to memorize, and have seen this question confound students in fifth and sixth grade classrooms, not because it requires a particularly long or difficult rule to memorize, but because they were unable to accept something as fact without knowing why (which is what makes a good mathematician!). Luckily, it’s a pretty easy concept to get your head around, no matter what age you are, if you’re familiar with a few basic rules of exponents.

Sample Problem

Why does 2^0 = 1, and not zero?

Solution

First, consider the following table of equations:

2^5 = 2x2x2x2x2 = 32
2^4 = 2x2x2x2 = 16
2^3 = 2x2x2 = 8
2^2 = 2×2 = 4
2^1 = 2 = 2
2^0 = ?????????????

In order to answer this question, first note that 32 = 2 x 16; therefore, we can say that the following equation is true: 2^5 = x^4 x 2. If we now divide both sides of the equation by 2, we can show that the following equation is also true: 2^5 ÷ 2 = 2^4.

Now, consider the following table of equations:

2^4 = (2^5) ÷ 2 = 32 ÷ 2 = 16
2^3 = (2^4) ÷ 2 = 16 ÷ 2 = 8
2^2 = (2^3) ÷ 2 = 8 ÷ 2 = 4
2^1 = (2^2) ÷ 2 = 4 ÷ 2 = 2

Finally, see what happens when we follow the same logical (and mathematically sound) approach to finishing this table:

2^0 = (2^1) ÷ 2 = 2 ÷ 2 = 1

Two to the power of zero can not equal zero, because it equals two divided by two. We (should) know that any number divided by itself is one, therefore 2 ÷ 2 = 1, so 2 ÷ 2 ≠ 0.



About The Author

Discrete Math And Conceptual Math Specialist
Hello, my name is Malachi. I am currently a graduate student in Mathematics Education at a University in northeast Oklahoma. I've had some great math teachers in my time as a student, and I've had at least as many poor math teachers. In honesty, this is one of the reasons I've pursued education in m...
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