Why any number with zero power is equal to 1?

Answer 1

Let a be any number and n any positive integer, an/an = 1 since since anything divided by itself is 1. But the laws of exponents say that an/an =an-n=a0 So this proves that a0 =1

Answer 2

0 to 0 is not defined-- there is no reasonable way to give it a meaning. However, the zero power of any non-zero number is defined to be 1. This turns out to be the correct definition because all theorems about powers and exponents hold true with this definition.

Another answer

Actually 0 to the power 0 can be taken to be 1. It can be interpreted as the number of functions from the empty set to the empty set: there is one such function (namely the empty set of ordered pairs).

A general comment is that this sort of definition isn't absolute. There is a bit of choice, but over time mathematicians settle on whichever definition works best in practice. As the first answer says, taking any number to be raised to the power 0 to give 1 means that theorems such as a^(b+c) = a^b x a^c work even if b = 0. (Here a^b means "a to the power b".)

Not everything can be defined. There is no reasonable way to define 0/0, because if we do give it a definite value we get into contradictions. But setting 0^0 = 1 doesn't give problems.

Answer 3

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.

Answer 4

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Look at nm / nm= nm-m=n0 but nm / nm=1 so we conclude that n0=1

So in particular 100=1

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