I often find that students do not know why 10 raised to the zero power equals one, i.e., 10^0=1. Frequently they are either completely at a loss or tell me that a teacher told them that it is just defined that way.

After all, we all know that ten squared means multiply 10 by itself two times to get 100, 10^3 means multiply ten by itself three times to get 1000, etc.

How can one multiply 10 by itself zero times and get 1 ??? Think about it! The answer follows below!

Why is 10^0 = 1?

As stated above, the meaning of ten to a positive power is easy to understand, e.g., 10^5 = 10x10x10x10x10 = 100,000. But how can one multiply ten by itself zero times and get 1 ?? We will get to the answer to this question by reversing the process and looking at division!

If I divide 100,000 by 1,000 I get 100 or, using exponents, 10^5 / 10^3 = 10^2 = 10^(5-3). Note that when we divide we subtract the exponents. This is because we have 5 tens in the numerator and three in the denominator, and the three in the denominator “cancel” (i.e., divide to a result of 1) three of the 5 tens in the numerator, leaving only 2 tens (or 10^2) in the numerator.

What happens if we divide 1000 by 1000? We clearly get 1 for the answer, but if we do this using exponents we get 10^3 / 10^3 = 10^(3-3) = 10^0 = 1! In effect, there are zero tens present in the answer, but a one remains after the division, not a zero!

In the following table the pattern is clear, and not only explains why 10^0 is 1, but also explains the meaning of negative exponents. **Note from each row to the next we divide by 10 or 10^1 on both sides of the equation.** On the left side when dividing by 10^1 we subtract the exponents, e.g. 10^4/10^1=10^(4-1)=10^3:

10^4 = 10000

10^3 = 1000

10^2 = 100

10^1 = 10

10^0 = 1

10^(-1) = 1/10 = 1/(10^1)

10^(-2) = 1/100 = 1/(10^2)

10^(-3) = 1/1000 = 1/(10^3)

etc.

In general 10^(-n) = 1/(10^n)

Finally, as long as we are on the mysteries of exponents, what is the meaning of a fractional exponent, e.g. 4^(1/2)?

Note that 4^(1/2) times 4^(1/2) = 4^((1/2)+(1/2)), because we add exponents when multiplying the same base (4 in this case). Clearly 4^(1/2) times 4^(1/2) = 4^1 = 4.

This means that if we multiply 4^(1/2) by itself we get 4. **That is the meaning of a square root!** 2 is the square root of 4 because if you multiply 2 times itself you get 4. Similarly 9^(1/2) * 9^(1/2) = 9^1 = 9 so 9^(1/2) is the square root of 9 or 3.

By similar logic 8^(1/3) * 8^(1/3) * 8^(1/3) = 8^(1/3 + 1/3 + 1/3) = 8 ^ 1 = 8. If you multiply 8^(1/3) by itself three times you get 8, so 8^(1/3) is the cube (third) root of 8 which is 2. 2*2*2=8.

By similar logic 10000^(1/4) is the fourth root of ten thousand which is 10.

Sadly when I talk about this with many high school students in precalculus and sometimes higher math, I often come away with the impression that the above explanation is completely new to them…

**PS – this leaves us with another tantalizing puzzle.**

Is it possible to have an irrational exponent like the number pi??

Does 2 ^ pi mean anything??

If pi was equal to 22/7, which I believe was decreed by law in the state of Indiana at one time, then 2^pi = 2^(22/7) = seventh root of 2^22.

This is an operation we could carry out on a calculator, but pi is a never ending decimal with no pattern and cannot be expressed as a fraction.

“Irrational” numbers are called irrational, not because they are crazy, but because they can not be expressed as ratios, i.e., fractions!

Is it even possible to compute 2 ^ pi ???

I’ll let you look that one up if you are interested!