The zero exponent rule says that any non-zero value raised to the zeroth power equals $\hspace{0.2em} 1 \hspace{0.2em}$.

In this tutorial, we'll look at one of the several exponent rules that help us work with and simplify exponents — the zero exponent rule.

The zero exponent rule says that any non-zero value raised to the zeroth power equals $\hspace{0.2em} 1 \hspace{0.2em}$.

$a^0 \hspace{0.2em} = \hspace{0.2em} 1, \hspace{0.5em} a \neq 0$

For example —

- $\hspace{0.2em} 9^0 = 1 \hspace{0.2em}$
- $\hspace{0.2em} 2.4^0 = 1 \hspace{0.2em}$
- $\hspace{0.2em} 1568^0 = 1 \hspace{0.2em}$

Of course, we can have algebraic expressions too as the base. But then, those expressions cannot themselves be $\hspace{0.2em} 0 \hspace{0.2em}$ in value.

Why can the base not be $\hspace{0.2em} 0 \hspace{0.2em}$, you ask? We'll get to that in a moment.

For example, $\hspace{0.2em} (x - 2)^0 = 1 \hspace{0.2em}$, given $\hspace{0.2em} x - 2 \neq 0 \hspace{0.2em}$ or $\hspace{0.2em} x \neq 2 \hspace{0.2em}$.

Similarly, $\hspace{0.2em} (2y + 7)^0 = 1 \hspace{0.2em}$, as long as $\hspace{0.2em} 2y + 7 \neq 0 \hspace{0.2em}$ or $\hspace{0.2em} y \neq -7/2 \hspace{0.2em}$.

Here's one simple explanation of why the zeroth power of any non-zero value equals $\hspace{0.2em} 1 \hspace{0.2em}$.

The quotient rule for exponents tell us —

$\frac{a^m}{a^n} \hspace{0.25em} = \hspace{0.25em} a^{m - n}$

Now, let's see what happens if we replace $\hspace{0.2em} 0 \hspace{0.2em}$ with $\hspace{0.2em} m - m \hspace{0.2em}$ in $\hspace{0.2em} a^0 \hspace{0.2em}$.

$\begin{align*} a^0 \hspace{0.25em} &= \hspace{0.25em} a^{m - m} \\[1.5em] &= \hspace{0.25em} \frac{\cancel{a^m}}{\cancel{a^m}} \\[1.5em] &= \hspace{0.25em} 1 \end{align*}$

See why $\hspace{0.2em} a^m = 1 \hspace{0.2em}$? Now, of course, $\hspace{0.2em} a \hspace{0.2em}$ can't be $\hspace{0.2em} 0 \hspace{0.2em}$ because in the second step, we are dividing by $\hspace{0.2em} a^m \hspace{0.2em}$. And we can't divide by $\hspace{0.2em} 0 \hspace{0.2em}$.

When it comes to $\hspace{0.2em} 0^0 \hspace{0.2em}$, things get a little murky. There is no concensus on what its value should be.

The majority of mathematicians and scientists seem to be of the opinion that the value of $\hspace{0.2em} 0^0 \hspace{0.2em}$ should be $\hspace{0.2em} 1 \hspace{0.2em}$, just like it is for the zeroth power of non-zero values.

But there are also many who think $\hspace{0.2em} 0^0 \hspace{0.2em}$ has an indeterminate value. Some even say it should be equal to $\hspace{0.2em} 0 \hspace{0.2em}$.

If you are just getting started with exponents, you need not worry about $\hspace{0.2em} 0^0 \hspace{0.2em}$ much. That said, in most case assuming its value is $\hspace{0.2em} 1 \hspace{0.2em}$ should work just fine.

Let me explain briefly what makes the case of $\hspace{0.2em} 0^0 \hspace{0.2em}$so special.

We have two different exponent rules that give two different values for $\hspace{0.2em} 0^0$.

One, $\hspace{0.2em} 0 \hspace{0.2em}$ raised to any non-zero exponent is $\hspace{0.2em} 0 \hspace{0.2em}$.

$0^n \hspace{0.2em} = \hspace{0.2em} 0, \hspace{0.5em} n \neq 0$

If we extend this rule to $\hspace{0.2em} 0^0$, it's value would be $\hspace{0.2em} 0 \hspace{0.2em}$.

Two, the zero exponent rule. Any non-zero base raised to the zeroth power is $\hspace{0.2em} 0 \hspace{0.2em}$.

$a^0 \hspace{0.2em} = \hspace{0.2em} 1, \hspace{0.5em} a \neq 0$

Now, if we extend this rule to $\hspace{0.2em} 0^0$, it's value would be $\hspace{0.2em} 1 \hspace{0.2em}$.

So you see, which of the two values do we go with — $\hspace{0.2em} 0 \hspace{0.2em}$ or $\hspace{0.2em} 1 \hspace{0.2em}$? Now, most people side with $\hspace{0.2em} 1 \hspace{0.2em}$ because the second rule is more consequential. And, hence, more important to remain consistent with.

And with that, we come to the end of this elementary tutorial on absolute value. Until next time.

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