0法则

Zero Exponent Rule

In other words, if you raise a nonzero number to the power of $0,$ the result is $1.$

Mathematicians debate the value of $0^0.$ Some say it's $1,$ and some say it's undefined.

Here are some examples of the Zero Power Rule. Notice it works for numbers and for variables.

$$\begin{align} 1^0&=1\\ (153+x)^0&=1, \quad\text{ if } x\neq -153\\ (x+y+z)^0&=1, \quad\text{ if } x+y+z \neq 0 \end{align}$$

So when $a \neq 0,$ why does $a^0=1?$ Why does the Zero Power Rule hold? Here's one way to think about it: $$a^0=a^{n-n}$$ Then, by the quotient rule for exponents, we can write this as: $$a^{n-n}=\frac{a^n}{a^n}$$

Then this becomes a problem about dividing fractions. Since the numerator and the denominator are both the same this becomes $1$. $$\frac{a^n}{a^n}=1$$

Determining The Zero Exponent

In mathematics, you try to make everything as consistent as possible. If a function works for positive numbers, how might it work for negative numbers? For zero?

Let's extend the exponent function from the positives down to $0$.

If we list the powers of $2$, for example, down to $0$, we get this pattern: $$\cdots 2^4,2^3,2^2,2^1,2^0\rightarrow \cdots 16,8,4,2,???$$

It appears that next number should be $1$, since that each number is the previous one divided by $2$.

If we try this for more numbers, we get the same pattern. All of them result in $1$ (except for $0$, but that is a different story). So we might propose a rule: $$x^0=1, \text{if } x\neq 0$$

A More Rigorous Proof of the Zero Exponent

Now we must find a way to confirm our finding.

We use an exponent identity. This one is called the Product Rule.

$$x^{m+n}=x^mx^n$$ $$x^{1+0}=x^{1}x^{0}$$ $$x^1=x^1x^0$$ $$\frac{x^1}{x^1}=\frac{x^1x^0}{x^1}$$ $$1=x^0$$

So we have proved our finding! The only caveat is when $x=0$; then, in the last step, we will be dividing both sides by $0$.

$0^0$

We left a gaping hole in our findings: what is $0^0$ exactly?

One argument could be that we might be able to arbitrarily extend the rule $x^0=1$ to $0$, thus making $0^0=1$.

But another argument goes like this: since $0^n=0$ for any nonzero $n$, we should extend that rule to $0^0$, thus making $0^0=0$.

Mathematicians eventually decided that such an argument was undecidable, so they decided to leave $0^0$ $\text{undefined}$.

(Video) Quotient Properties and Negative Exponents

by ThinkwellVids

This video by ThinkwellVids explains what the theory behind numbers raised to a power of $0$.

$\large{\text{Summary}}$

Raising any number to zero gives you $1$ as an answer. In other words: $$a^0=1\quad\text{where }x\ne0$$

Basically, if you have $\frac{a}{a}$ this equals $1$. You can raise this to any power you want,

$$\left(\frac{a}{a}\right)^m=\frac{a^m}{a^m}$$

The rule for dividing exponents says that $\frac{a^m}{a^m}=a^{m-m}=a^0$.

$$1=\frac{a}{a}=\left(\frac{a}{a}\right)^m=\frac{a^m}{a^m}=a^{m-m}=a^0$$