Zero Exponent Rule

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.

Zero Exponent Rule

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

a0=1,a0a^0 \hspace{0.2em} = \hspace{0.2em} 1, \hspace{0.5em} a \neq 0

For example —

  • 90=1\hspace{0.2em} 9^0 = 1 \hspace{0.2em}
  • 2.40=1\hspace{0.2em} 2.4^0 = 1 \hspace{0.2em}
  • 15680=1\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 0\hspace{0.2em} 0 \hspace{0.2em} in value.

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

For example, (x2)0=1\hspace{0.2em} (x - 2)^0 = 1 \hspace{0.2em}, given x20\hspace{0.2em} x - 2 \neq 0 \hspace{0.2em} or x2\hspace{0.2em} x \neq 2 \hspace{0.2em}.

Similarly, (2y+7)0=1\hspace{0.2em} (2y + 7)^0 = 1 \hspace{0.2em}, as long as 2y+70\hspace{0.2em} 2y + 7 \neq 0 \hspace{0.2em} or y7/2\hspace{0.2em} y \neq -7/2 \hspace{0.2em}.

Making Sense of the Zero Exponent Rule

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

The quotient rule for exponents tell us —

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

Now, let's see what happens if we replace 0\hspace{0.2em} 0 \hspace{0.2em} with mm\hspace{0.2em} m - m \hspace{0.2em} in a0\hspace{0.2em} a^0 \hspace{0.2em}.

a0=amm=amam=1\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 am=1\hspace{0.2em} a^m = 1 \hspace{0.2em}? Now, of course, a\hspace{0.2em} a \hspace{0.2em} can't be 0\hspace{0.2em} 0 \hspace{0.2em} because in the second step, we are dividing by am\hspace{0.2em} a^m \hspace{0.2em}. And we can't divide by 0\hspace{0.2em} 0 \hspace{0.2em}.

What About 00\hspace{0.2em} 0^0 \hspace{0.2em}?

When it comes to 00\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 00\hspace{0.2em} 0^0 \hspace{0.2em} should be 1\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 00\hspace{0.2em} 0^0 \hspace{0.2em} has an indeterminate value. Some even say it should be equal to 0\hspace{0.2em} 0 \hspace{0.2em}.

If you are just getting started with exponents, you need not worry about 00\hspace{0.2em} 0^0 \hspace{0.2em} much. That said, in most case assuming its value is 1\hspace{0.2em} 1 \hspace{0.2em} should work just fine.

Why So Peculiar?

Let me explain briefly what makes the case of 00\hspace{0.2em} 0^0 \hspace{0.2em}so special.

We have two different exponent rules that give two different values for 00\hspace{0.2em} 0^0.

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

0n=0,n00^n \hspace{0.2em} = \hspace{0.2em} 0, \hspace{0.5em} n \neq 0

If we extend this rule to 00\hspace{0.2em} 0^0, it's value would be 0\hspace{0.2em} 0 \hspace{0.2em}.

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

a0=1,a0a^0 \hspace{0.2em} = \hspace{0.2em} 1, \hspace{0.5em} a \neq 0

Now, if we extend this rule to 00\hspace{0.2em} 0^0, it's value would be 1\hspace{0.2em} 1 \hspace{0.2em}.

So you see, which of the two values do we go with — 0\hspace{0.2em} 0 \hspace{0.2em} or 1\hspace{0.2em} 1 \hspace{0.2em}? Now, most people side with 1\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.