Indices rule for division with 0

Question

The indices rule for division states

$ m^x \div m^y = m^{x-y}. $

This works out because

$ m^4 \div m^2 = {{m*m*m*m} \over {m*m}} = m*m = m^2 = m^{4-2} $

So we see that through the reduction rule for division, $m^y$ cancels out.

In my mind, this should break when $m=0$, since you cannot divide by $0$. But any $0^x = 0$ as long as $x\ne0$.

So, for example, what is the solution to the following?

$ 0^4 \div 0^2$

Is it $0$ or undefined?

Answer 1

That actually depends on what you’re looking at. If you simply look at the expression $0^m/0^n$ then this is not defined, as $0$ is not invertible. But if you actually look at polynomials $P/Q$ that both have a root of order $m$/$n$ in some point $x$ you can in fact do polynomial division to make sense of this.

Answer 2

Any time you have a rule like this, there is implicitly a domain in which it can be applied and in which it makes sense.

In this case, since division is involved, the domain is restricted by $m \neq 0$.

Answer 3

Note that $0^4 \div 0^2$ is the same as $0 \div 0$, so let's focus on that.

(Answer) The expression $\frac{0}{0}$ is undefined. The fraction $\frac{a}{b}$ means "the number which when multiplied by $b$ gives $a$." For example, the reason $\frac{1}{0}$ is undefined is because there's no number $c$ where $0 \cdot c = 1$.

The case for $\frac{0}{0}$ is weird at first because all numbers $c$ satisfy $0\cdot c = 0$. But since there's no single $c$ that works (in fact, infinitely many choices of $c$ would work), there's no clear way to define $\frac{0}{0}$. Therefore by convention, we leave it undefined.

Does that answer your question?