Raising to the zero and negative powers

Task

In this problem $c$ represents a positive number. 

The quotient rule for exponents says that if $m$ and $n$ are positive integers with $m>n$, then $$\frac{c^m}{c^n} = c^{m-n}.$$ After explaining to yourself why this is true, complete the following exploration of the quotient rule when $m\leq n$:

1. What expression does the quotient rule provide for $\frac{c^m}{c^n}$ when $m =n$?
2. If $m = n$, simplify $\frac{c^m}{c^n}$ without using the quotient rule.
3. What do parts (a) and (b) above suggest is a good definition for $c^0$?
4. What expression does the quotient rule provide for $\frac{c^0}{c^n}$?
5. What expression do we get for $\frac{c^0}{c^n}$ if we use the value for $c^0$ found in part (c)?
6. Using parts (d) and (e), propose a definition for the expression $c^{-n}$.

IM Commentary

The goal of this task is to use the quotient rule of exponents to help explain how to define the expressions $c^k$ for $c \gt 0$ and $k \leq 0$. This important definition is motivated and explained by the law of exponents: adopting the definitions for the expressions $c^0$ and $c^{-n}$ given in the task allows us to maintain the intuitive product and quotient rules known for all positive exponents (which this task assumes students are familiar with).

The standard 8.EE.1 calls for knowing and applying properties of integer exponents. In order to do this, students need to understand why $c^0 = 1$ and also how make sense of $c^k$ when $k$ is a negative integer. For positive integers, students know that $c^{m} \times c^n = c^{m+n}$ by the meaning of these exponential expressions. This important relationship still holds when $m$ and $n$ are integers but this needs to be established. This task is a first step toward this goal. It provides a way of making sense of integer powers of a positive number but does not check that the product rule of exponents holds for these more general expressions.

It is important in the task statement that $c$ is non-zero. The expression $0^0$ is undefined and much more is written about this in the following high school task: https://www.illustrativemathematics.org/illustrations/1823. While it does make sense for $c$ to be negative, this can lead to issues when the exponent is a fraction. This task does not examine the quotient rule when one or both of the exponents is negative. The teacher may wish to have students explore these cases as an extension of the present problem.

Solution

1. If we apply the quotient rule for exponents when $m = n$, we find \begin{align}\frac{c^m}{c^n} &= \frac{c^m}{c^m} \\ &=c^{m-m}\\ &= c^0.\end{align}

2. Without the quotient rule, when $m = n$, we have $\frac{c^m}{c^m} = 1$. 

3. Parts (a) and (b) give a compelling justification for choosing to define $c^0=1$ for any positive number $c$.  Namely, on the one hand, $\frac{c^m}{c^m}$ is 1 by doing the explicit division. On the other hand, the extension of the quotient rule to the case $m=n$ gives $\frac{c^m}{c^m}=c^0$.  If we want the quotient rule to continue hold in this case, we are forced to define $c^0=1$.

4. If we apply the quotient rule for exponents when $m = 0$, we find \begin{align}\frac{c^0}{c^n} &= c^{0-n}\\ &= c^{-n}.\end{align} We have a negative exponent in this case.

5. According to part (c), $c^0 = 1$. If we substitute this into the fraction $\frac{c^0}{c^n}$ we find that $\frac{c^0}{c^n} = \frac{1}{c^n}$.

6. Much like our definition for $c^0$, the previous two parts dictate that if we want the quotient rule to hold for the expression $\frac{c^0}{c^n}$, we are forced to define $c^{-n} = \frac{1}{c^n}$. That is, the quotient rule suggests we should define raising the number $c$ to a negative power to be the reciprocal of the corresponding positive power.