# How should I convince a student who thinks they proved $1=-1$

One of my high school students who has ZERO knowledge on complex numbers and the modulus function has showed me the following algebra:

$$(16)^{\frac{1}{2}}=(16)^{\frac{2}{4}}=((16)^2)^{\frac{1}{4}}=((-16)^2)^{\frac{1}{4}}=(-16)^{\frac{2}{4}}=(-16)^{\frac{1}{2}}$$

Hence $$16=-16 \qquad\text{and so}\qquad 1=-1.$$

Should we impose that $$(a^m)^n=a^{mn}$$ only when $$a \gt 0$$?

• You may be interested in this question from MESE's sister site, MSE. Commented Feb 24, 2020 at 16:01

Should we impose that $$(a^m)^n=a^{mn}$$ only when $$a \gt 0$$?

Maybe you should tell your student that he/she have discovered by himself/herself the proof that the rule $$(a^m)^n=a^{mn}$$ cannot be true with negative bases and rational exponents, and this is a great achievement.

Try to explain that he/she proved that

If the said rule is true for negative bases and rational exponents, then $$1=-1$$.

As we know that $$1\neq -1$$, the rule cannot be true. This is why the restriction $$a>0$$ appears in the books and why you will impose it.

## Edit

• Maybe I'm wrong but, in view of the title's question ("How should I convince a student..."), I suppose that the question is not about the rule $$(a^m)^n=a^{mn}$$ itself (which, I believe, the OP knows), but on how to approach it with the student.

• The essence of my suggestion is: tell to the student what he/she did right (a mathematical discovery) instead of focusing on the mistake (wrong application of a rule that, probably, the student didn't know yet).

• Of course, the bold text above is not as general as possible. But this is only the first step (see my comments below). Afater that, the instructor could guide the student to further investigations in order to discover other forms of the rule.

• Note that most high school/college algebra textbooks I'm aware of do define values of $a^{m/n}$ with $a < 0$ and odd $n$. Commented Feb 25, 2020 at 18:21
• @DanielR.Collins Then, the restriction mentioned in my post have to be replaced by the appropriate one. Commented Feb 25, 2020 at 18:34
• @DanielR.Collins Did you realize that the question isn't about the most general condition under which the rule $(a^n)^m=a^{nm}$ is valid? Commented Mar 6, 2020 at 14:06
• @pedro Maybe the rest of the class wouldn't care, but I think this precocious student would be unsatisfied with a simple prohibition on negative bases or that they are somehow "beyond the scope" of the course. See my suggestion in my new answer to give him/her a bit more to "chew on." You could assure the rest of the class that they will be tested only on positive bases if they get restless. Commented Mar 8, 2020 at 7:00
• @DanChristensen "a simple prohibition" is exactly the opposite of my suggestion. If you didn't realize it, you didn't understand it. On the other hand, isn't "don't do that if $a^m\notin R$ or $a^n\notin R$" a simple prohibition? The student's reasoning has its value. It's this reasoning that leads us to the correct form of the rule. This perspective is completely lost in your suggestion. Commented Mar 8, 2020 at 13:04

Obviously the correct mathematical answer is to show how the exponent rules actually work, and when they do not work.

Anyway, the educational answer is to see that student is using the fact that $$1^2 = (-1)^2$$ in the critical middle step. Show them the corresponding fact for cubes, because it's crazy, and might expand their mathematical playground:

$$1^3 = \bigg( \frac{-1 + i\sqrt{3}}{2} \bigg)^3$$

See what they can "prove" next. your student has an imagination; show them the rules and when they work -- but also keep that spark alive by showing them neat things.

When I asked the What are the Laws of Rational Exponents? question on SE Mathematics, I was largely thinking about this context; teaching at the level of high school or early (remedial) college math. While it wasn't the top-voted, my answer there represents my best thinking about the status of this issue in classes at that level. As I wrote:

Regarding the example in the question, most everyone agrees that $$(-1)^{2 \cdot \frac{1}{2}} \ne ((-1)^2)^\frac{1}{2}$$, if both sides are simplified in the standard order of operations; and this highlights the fact that the identity $$(a^r)^s$$ = $$a^{rs}$$ is not true unrestrictedly. Exactly what restrictions need to be honored depend on the definitions in use in a particular textbook.

Just to expand on the last line there; part of the conversation around these questions usually dredges up the fact that there is a standard definition for fractional exponents in real-numbers, and another one in complex-numbers, and that these two definitions actually disagree with each other (e.g.: the value of $$(-8)^{1/3}$$, on which numerous academic articles have been written). As a result, real-number theorists and complex-numbers theorists have a tendency to start arguing with each other on these questions.

So: I assume you are working from some standard textbook. That textbook must have some restrictions around the $$(a^m)^n = a^{mn}$$ identity (although they vary between books and contexts). I would recommend you read your book very carefully, note the restriction used there, and apply it conscientiously in your classroom practice.

• I can't help but express hesitation about "definitions used in a given book" as a way to explain mathematical facts... :) Commented Apr 25 at 19:28
• I'm aware that this answer is a few years old, but as this just popped up at the front page I'm wondering what a "real-number theorist" is. Maybe this refers to real analysts? But I'm not sure why a real analyst would argue about this issue - I've never seen fractional powers of negative numbers used in real analysis except for the case where the exponent is of the particular form $1/n$ for odd $n$ (and I'm also wondering for which purpose one would want to define them for more general exponents). Commented Apr 25 at 19:59

I think the previous answers have focused too much on the details of rational exponents and negative bases. There is a much simpler point about logic that resolves this whole example and that I think would be much more appropriate as an answer to a student.

I have a dog who has a two eyes. Your mother also has two eyes. Does this imply that your mother is a dog?

No. We have a method or rule such that, given a physical being as an input, we can count eyes and find an output. Equality of the outputs does not imply equality of inputs.

So even if it were true that $$1^{1/2}=(-1)^{1/2}$$, that would not have the consequence that $$1=-1$$.

• I don't know why this was downvoted. Maybe it doesn't resolve the whole example, but it's a good point. If the idea is to have the student analyze basic rules of math, surely pointing out that not every function is injective is helpful. Commented Apr 25 at 21:40

(My new and hopefully improved answer)

Should we require that $$(a^m)^n=a^{mn}$$ only when $$a \gt 0$$ ?

That might "solve" the problem in some sense, but we do have legitimate cases with negative bases.

Example: $$((-2)^2)^3 =(-2)^{2\times 3}=(-2)^6=64$$

According user "Gae. S." at Math SE, $$a^m, a^n \in \mathbb{R}$$ is a sufficient condition for $$(a^m)^n=a^{mn}$$ when using real-number arithmetic.

So the Power of a Power Rule (on $$\mathbb{R}$$) can be restated: $$(a^m)^n = a^{mn}$$ if we have $$a^m, a^n \in \mathbb{R}$$.

This may provide a more satisfying resolution for this precocious student: Since $$(-16)^{1/4} \notin \mathbb{R}$$, we cannot infer that $$((-16)^2)^{\frac{1}{4}}=(-16)^{\frac{1}{2}}$$.