Why does any number to the zero power equal one?

Answer 1

Here is why any number to the zero power equals one.

Consider this.

a^b. it is natural to restrict a > 0, but we'll only assume that number b is any real number.

We'll use the natural exponential function defined by the derivative of the exponential function.

Now we have a^r=e^rln(a). And we know that e^rln(a)=e^((ln(a))^r), where a >0 and r is in the domain of all real numbers negative infinity to infinity.

We can apply this definition to any number a to any power r.

Particularly, a^0. By the provided definition, a^0=e^(0*ln(a))=e^0=1.

Furthermore, a^1=e^(1*ln(a))=e(ln(a))=a.

And a^2=(e^(ln(a))^2)=a^2.

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Here is a simpler approach:

In general, a^n/a^m = a^(n-m) and a/a = 1. We can use these facts to prove that x^0 = 1 so long as x isn't 0.

First, state the obvious: 1 = 1

Next, since any non-zero number divided by itself is one: 1 = a^n/a^n

(It doesn't change how the equation looks, but for the sake of being thorough, you could subsitute (a^n/a^n) in place of 1 in the original equation.)

Then, since dividing like bases requires that you subtract their exponents:

a^n/a^n = a^(n-n) = a^0

Substitute (a^0) in for (a^n/a^n) and you obtain: a^0 = 1

There are two reasons "a" cannot be 0 in this proof: firstly, raising 0 to non-zero powers would still result in zero, so "a" being 0 would cause division by zero in the initial theorems we used, and secondly, 0^0 is considered undefined in itself.

Answer 2

If you think what raising to a power means:- n to power 2 is n multiplied by itself once,

n to power 3 is n multiplied by itself twice . In general n to the power x = n multiplied by itself (x-1) times so n to the power 1 will be n multiplied by itself 0 times = n (not the same as multiplying by zero - just not multiplying n by anything). The big leap then is too see that n to the power zero is n multiplied by itself minus 1 times which is n divided by n =1.

To see this more clearly picture the series created by any whole number raised to a power increasing by one incrementally from left to right.

...................n, nxn, nxnxn, nxnxnxn, nxnxnxnxn, ....................

each new term is the old term multiplied by n. So if you go in the opposite direction each new term is the old term divided by n so:-

n to power 2 divided by n =n to power 1 which is n so n to power 0 is n divided by n which is one, continuing on n to the power -1 is 1 divided by n (one nth) and n to the power -2 is 1 divided by n squared.