Why is 10^0 = 1 ???

A common question I ask my students – their answers make me question the quality of their math education…

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!

Author: David Kristofferson

Retired scientist, teacher, bioinformatician, IT director, software product manager, AAAS Fellow, avid cyclist (7690 miles and 724,300 feet of climbing in 2015), backpacker, you name it! Current avocation is tutoring high school students near San Mateo, CA in mathematics, physics and chemistry. Please see the Bio link in the right sidebar for my detailed background information.

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