Exponents of the Zero Power

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Someone wrote in to ask:

4⁰ * 5³ is 125. Why isn’t it 0?

On the Math.Com website, problems such as 4 to the zero power times 5 to the third power have an answer of 125 as correct. Shouldn’t the answer be zero. If not, why? Thank you!

Professor Homunculus’ response:

The answer actually should not be zero, and here’s why:

Because 4 to the 0 power is 1, not 0.

So 4⁰ * 5³ would be 1 x 5³ which is 125.

Any integer raised to the zero power equals 1.

That is hard for most people to believe, so I wrote a little piece to explain why it makes sense. Here it is:

In modern mathematics, we usually use a base system to represent our numbers.

You know that we have units, tens, hundreds, etc. in our base-ten system.
Well, we also represent those columns in terms of 10 to the nth power. In other words, the thousands column is represented by 10³. So 8,000, for example is 8 *10³.

• 8,300 would be (8 * 10³) + (3 * 10²).
• 8,320 would be (8 * 10³) + (3 * 10²) + (2 * 10¹).

Now comes the problem. You see how the base stays the same, and the exponent gets smaller? To represent the units column, mathematicians have accepted the convention that 10⁰ will always equal 1.

That keeps the rule going. So 8,325 would be:
(8 * 10³) + (3 * 10²) + (2 * 10¹) +(5 * 10⁰).

Because other bases systems, (base 2, base 3, etc.) work the same way, we have further accepted the convention to be n⁰ always equals 1, no matter what the base.

Take the binary (base 2) number 1011 for example. What that means is
(1 * 2³) + (0 * 2²) + (1 * 2¹) + (1 * 2⁰).

That is the same as 8 + 0 + 2 + 1,

which is the number 11 in our normal base 10 system.

As long as we keep n⁰= 1, then the units column of any base will always mean how many ones there are in it.

Math is a network of "conventions" mankind has accepted to make it work. It is based on rules and axioms that are the most "convenient". They may seem hard to figure out at first, but when you get down to it, the rules that we use are basically the best we can come up with to get the things done that we want to accomplish.

Every once in awhile some genius comes up with a rule that makes something even simpler than how we have been doing it up until now, and that becomes the new convention. But that rule has to be solidly based on what has come before, and may not break any of the other rules.

Maybe someday you will be one of the people who comes up with something that explains, or enables something that until now was done by a convention that needed improving.

Have fun!

Professor Homunculus