How can $x^0=1$ be proved with a real life experiment?

Question

I saw that $x^0=1$ proof is $\displaystyle x^{a-a} = \frac{x^a}{x^a} = 1.$

However I don't find this a convincing proof.

How can someone prove this without using the fraction above? or, Is that fraction the only proof of $x^0=1$?

Answer 1

With compound interests, your capital increases by $10\%$ every year.

$$1 \to 1.1\\ 2 \to 1.21\\ 3 \to 1.331\\ 4 \to 1.4641\\ \cdots\\ N\to1.1^N \cdots$$

Obviously we get a consistent definition if

$$0\to1$$

Answer 2

If your notion of exponentiation is ultimately based on repeated multiplication (there are other ways to approach powers, where the answer may differ), then $x^0=1$ cannot be proven. It is defined to be such. And your fraction is one of many, many reasons that $1$ is the most reasonable value to give $x^0$, by a wide margin.

Answer 3

You can only prove mathematical statements with arguments that start from mathematical axioms. "Real life" does not enter the picture.

That said, the assertion that for $x>0$ $$ x^0 = 1 $$ is not a theorem, it's a definition. Mathematicians could have decided that $$ x^0 = 17, \text{ or } 0, \text{ or } \ldots $$ but chose $1$ because it turns out to be the most useful. You are entitled to an explanation for that choice.

Remember that for ordinary positive integers, $$ x^m x^n = x^{m+n} $$ (true because $x^n$ is defined as the result of multiplying $x$ by itself $n$ times).

That rule for adding exponents is so useful that mathematicians decided to preserve it when extending the meaning of raising $x$ to a power. Then $$ x^0 x^n= x^{0+n} = x^n . $$ The only way that can happen is if you define $x^0$ to be $1$.

The same desire to preserve the rule for adding exponents is why $$ x^{-1} = \frac{1}{x} \text{ and } x^{1/2} = \sqrt{x}. $$

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For another good reason to define the product of no numbers as $1$ see Empty set and empty sum

Answer 4

Start with the following equation

$$x=a, $$

in which $a>0$ and apply the $n^\text{th}$-root.

$$x^{\frac 1n}=a^ {\frac 1n}$$

If you take the limit $n \to \infty$ you will obtain

$$x^0=1,$$

because the infinith root of a positive number is $1$.