math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

Question

I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$

See http://math.stackexchange.com/questions/49169/i2-why-is-it-1-when-you-can-show-it-is-1 — Gamma Function, Aug 20 '13 at 19:02
Maybe someone can create sample general reference question for $$ \sqrt{x^2} = x $$ fallacy. Then we can just mark similar questions as dupes. — default locale, Aug 21 '13 at 08:40
I think people are being overly complicated in bringing up multivaluedness of $\sqrt{}$. It does play a role, but by interpreting everythingn in the normal way on the complex plane, mrf's solution is the simplest, and doesn't require excuses about nonprincipal square roots. — rschwieb, Aug 23 '13 at 16:49
I agree with @JacobMayle and "default locale". It can be considered as a duplicate. — vesszabo, Aug 27 '13 at 19:46
@defaultlocale I don't see this as assuming $\sqrt{x^2} = x$. The right-most equality is achieved by $\sqrt{\left(-1\right)^6} = \sqrt{1} = 1$, which is clearly valid. The suspect part is $\left(-1\right)^{6/2} = \left(\left(-1\right)^6\right)^{1/2} = \sqrt{\left(-1\right)^6}$ — Tim Goodman, Mar 20 '14 at 06:16
@TimGoodman what happens when you replace $(-1)^3$ with $x$ in the suspect part? — default locale, Mar 20 '14 at 06:58
@defaultlocale I don't follow. The expression $(-1)^3$ does not occur in what I referred to as the suspect part. Perhaps you are saying that the fallacious statement $(-1)^{6/2} = \sqrt{(-1)^6}$ implies the fallacious statment $\left(-1\right)^{6/2} = \sqrt{\left(\left(-1\right)^{6/2}\right)^2}$? And thus it implies $(-1)^3 = \sqrt{\left((-1)^3\right)^2}$? You can always get falsehood from falsehood, but that doesn't meant the resulting falsehood was where your original wrong assumption lies. — Tim Goodman, Mar 20 '14 at 07:09
@TimGoodman I don't think I understand. Let us continue this discussion in [chat](http://chat.stackexchange.com/rooms/13721/discussion-on-fallacy) — default locale, Mar 20 '14 at 07:17
http://math.stackexchange.com/questions/1395537/what-is-the-fallacy-of-this-proof — Soham, Mar 06 '16 at 11:57
related: https://math.stackexchange.com/questions/1628759/what-are-the-laws-of-rational-exponents/3362483#3362483 — Mike Earnest, Apr 10 '21 at 19:52

score 171 · Answer 1 · answered Aug 20 '13 at 18:45

171

The "rule" $(a^b)^c = a^{bc}$ doesn't necessarily hold when $a < 0$.

answered Aug 20 '13 at 18:45

mrf

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In my opinion this is the only answer that exactly says where the problem is. – Git Gud Aug 20 '13 at 18:48
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I would add that it holds when `b` and `c` are integers but not necessarily when one (or both of them) aren't. – ypercubeᵀᴹ Aug 20 '13 at 23:28
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@ypercube The second equality only uses that $6/2=3$. I don't see how anyone could think that's what I mean. – mrf Aug 21 '13 at 15:12
Wow, what?????? – bodacydo Aug 22 '13 at 00:29
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$(a^b)^c = a^{bc}$ doesn't hold when $a^b$ or $a^c$ have cuts in the complex plane. If we defined $sqrt(1) := -1$ the paradox presented in this question wouldn't hold. – jimifiki Aug 22 '13 at 06:56
I think that it should be noted that $\sqrt{a^2} = | a |$. So, when you calculate $\sqrt{(-1)^6}$ you are doing something like this: $\sqrt{(-1)^6}=((-1)^2)^\frac{3}{2}=\left(\sqrt{(-1)^2}\right)^3=(|-1|)^3=1$ – Barranka Aug 27 '13 at 20:49
@Barranka $(-1)^6=1$, so $\sqrt{(-1)^6}=1$. – mrf Aug 27 '13 at 20:51
Sir, what you say is wrong. 'a' irrespective of the sign, satisfies the property. (a^b)^c can be written as (a.a.a....b times)^c, which is the same as (a.a.a....b times)(a.a.a....b times).....c times, which is the same as a.a.......(bc) times. – Swapnil Tripathi Mar 26 '14 at 06:03
Where b and c are positive. – Swapnil Tripathi Mar 26 '14 at 06:05

score 50 · Answer 2 · edited Aug 21 '13 at 16:30

50

There is a simpler version of this fallacy: $-1 = (-1)^{2/2} = \sqrt{(-1)^2} = \sqrt{1} = 1$. The mistake comes from the fact that the function $f(x)=x^2$ is not invertible so you cannot conclude that for any real number $x$ it is the case that $x = \sqrt{x^2}$.

There is a version of the same mistake that uses the fact that $log$ is not invertible on $\mathbb{C}$ to prove that all numbers equal 1:

$x = e^{\ln(x)} = e^{\ln(x) * (2\pi i) / (2\pi i)} = (e^{2\pi i})^{\ln(x)/2\pi i} = (\cos(2\pi)+i \sin(2\pi))^{\ln(x) / 2\pi i} = 1^{\ln(x) / 2\pi i} = 1$

edited Aug 21 '13 at 16:30

obataku

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answered Aug 20 '13 at 18:48

exk

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I think the first part of your answer is correct. But for the second part, I think the problem is not that $\log$ is not invertible, but is still as the first answer says that $(a^b)^c=a^{bc}$ is not true for all $a$ in general. – L. Xu Aug 22 '13 at 19:58
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@L.Xu I disagree. The problem in OP's set of equalities is that $x=\sqrt{x^2}$ is true when $x^2$ is invertible, e.g., if we assume that $x$ is a positive real number. (To be precise, we have $x=\sqrt{x^2}$ if $x^2$ is invertible because we are working with strictly negative numbers; in that case, its inverse is $-\sqrt{\cdot}$. The problem in the fake proof that all numbers equal $1$ is that $x=e^{\ln(x)}$ is true when $ln(x)$ is invertible, e.g., if we are working in $\mathbb{R}$ but not if it is not, e..g, if we are working in $\mathbb{C}$. – exk Aug 23 '13 at 00:59
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One way to see that this is about invertability and not about the equality $(a^b)^{c}$ _per se_ is to see that the same mistake can creep in if you use non-invertable function other than $x^n$. Consider the following fallacy: $x=sin^{-1}(sin(x))=sin^{-1}(sin(2\pi x)=2\pi x$. Hence $\forall x, x=2\pi x$ which implies $\forall x, x=0$. – exk Aug 23 '13 at 01:07
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Dear @exk : The invertibility is tied together with the failure of $(a^b)^c=(a^c)^b$. If you interpret in complex numbers, $-1=(-1^{1/2})^2=i^2$, but $((-1)^2)^{1/2}=1$. In other words *the order you do the exponents matters when you have multiple valued roots.* I think you are *both* right :) – rschwieb Aug 23 '13 at 16:44

Bran · Answer 3 · 2013-08-22T16:28:05.530

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What is wrong here is assuming that $\sqrt{x^2} = x$ when the fact is $\sqrt{x^2} = |x|$. Let $x=−1$ and use $\sqrt{x^2} = |x|$ in the problem above, you should arrive at a valid equation.

edited Aug 22 '13 at 16:28

answered Aug 21 '13 at 11:31

Bran

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I'm surprised no one wrote that earlier @subzero...nice addition – Eleven-Eleven Aug 21 '13 at 11:58
Hear hear (they did say it, just not in so many word, which really is surprising). – Jonathan Y. Aug 21 '13 at 17:30
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What you say is correct, but how does this help clear up the OP's problem? – us2012 Aug 21 '13 at 20:56
@us2012 let $x=-1$ and use that $\sqrt{x^2}=|x|$ in the problem above, you should arrive at a correct equation. – Bran Aug 22 '13 at 10:08
Not clear to me how this relates to the original problem. Subzero, you write "let $x = -1$ and use that $\sqrt{x^{2}} = \vert x \vert$..." However, there IS NO $x$ int he original problem, so there is nothing that can be substituted. – Florian Jul 11 '17 at 09:53
@Florian The general case is what I meant – Bran Nov 17 '17 at 14:57

score 14 · Answer 4 · answered Aug 20 '13 at 18:49

14

As other say the square root may be two valued. However you use it as a function, so it's single valued. It really depends on your definition of used functions. I think that $(-1)^{6/2} = -1$ but $\sqrt{(-1)^6} = ((-1)^6)^{1/2} = 1$. So @mrf is right that $(a^b)^c ≠ a^{bc}$ in general and the third equality of your equation is the one that doesn't hold.

answered Aug 20 '13 at 18:49

user87690

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Important point mentioning there's no problem with taking the square root. – Git Gud Aug 20 '13 at 18:50

score 9 · Answer 5 · answered Aug 20 '13 at 18:44

9

The problem here is that the square root function, $\sqrt{-},(-)^\frac{1}{2}$, is not a single-valued function.

As PVAL says, it is a two-valued function, meaning you have two consistently choose which square root you're talking about. That's why you often will have problems when you have chains of equalities as above.

answered Aug 20 '13 at 18:44

Fredrik Meyer

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The function $x \mapsto \sqrt{x}$ *is* single valued for all $x \ge 0$. *By definition* $\sqrt{2}$ is the *positive* solution to the equation $y^2 = 2$. That is why we write $\pm \sqrt{2}$. It is the function $x \mapsto x^{1/2}$ which is multi-valued. – Fly by Night Aug 20 '13 at 18:58
@FlybyNight I'm not sure many people make this distinction between $\sqrt{x}$ and $x^{1/2}$. The reasoning in the OP breaks down even when we think only about single valued functions, here. – rschwieb Aug 23 '13 at 16:56
@rschwieb But they *should*... For example $3^{1/2} = \pm\sqrt{3}$ and not just $\sqrt{3}$. Similarly $1^{1/4} = \pm 1, \pm\operatorname{i}$ and not just $\sqrt[4]{1} = 1$. According to Wikipedia "*Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by $\sqrt{a}$, where $\sqrt{}$ is called the radical sign or radix.*" – Fly by Night Aug 23 '13 at 17:51
Dear @FlybyNight : *Maybe*, but I'm reluctant to moralize about it. This is more likely to confuse readers at a basic level rather than help. Maybe a little less so now, since we've discussed it directly :) – rschwieb Aug 23 '13 at 18:11
@rschwieb I don't think it will confuse them. It is just high-school algebra. What is $x^{1/2}$? Well, assuming $x>0$ then $x^{1/2} \times x^{1/2} = x^1 = x$. So $x^{1/2}$ is a number that when you multiply it by itself you get $x$. Since $(-2) \times (-2) = +4$ and $(+2) \times (+2) = +4$ we can see that $4^{1/2}$ can be *either* $-2$ or $+2$. – Fly by Night Aug 23 '13 at 18:24
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Dear @FlybyNight I do not believe that multifunctions are standard high school fare (at least in the US) nor do I think they should be as long as students continue to struggle with single-valued functions (I'll moralize on that :) ). You needn't explain details of your argument: I'm not faulting it anywhere! I just think it's not as accessable a reason. – rschwieb Aug 23 '13 at 18:29
@rschwieb The fact that $x^a \times x^b = x^{a+b}$, for a ppropriate $x$, is high-school algebra. The fact that the solution to $y^2=4$ is either $y=-2$ or $y=+2$ is high-school algebra. My last comment uses nothing more than these two facts. – Fly by Night Aug 23 '13 at 18:57

score 6 · Answer 6 · answered Aug 20 '13 at 18:43

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Taking square roots is in a sense a two valued function, because every non-zero complex number $z$ has two distinct complex numbers $w_1, w_2$ for which $w_1^2=w_2^2=z$.

answered Aug 20 '13 at 18:43

walcher

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score 5 · Answer 7 · answered Aug 21 '13 at 17:37

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$\sqrt{x^2} = +x$ or $-x$

The fault with the "proof" is the false assumption that you can choose the positive root and still have everything hold. In fact, you need to select the correct root based on context or accept two possible answers.

answered Aug 21 '13 at 17:37

Jack Aidley

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Swapnil Tripathi · Answer 8 · 2014-10-18T17:38:36.650

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You say, $(-1)^{6/2}= [(-1)^6]^{1/2}$

I'd rather see it as, $(-1)^{6/2}=[(-1)^{1/2}]^6$

which is equal to $i^6$ or $(-i)^6 = -1$

(Just for fun : $DMAS$ rule for powers? :P)

edited Oct 18 '14 at 17:38

answered Mar 25 '14 at 19:32

Swapnil Tripathi

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Jonathan Viccary · Answer 9 · 2015-01-25T22:16:40.320

The correct use of ${\sqrt{}}$ in this context would be

$$ −1 = (−1)^3 = (−1)^{6 / 2} = -\sqrt{(-1)^6} = -\sqrt{1} = -1$$

and this is simply a consequence of the inverse of $x^2$ being $-\sqrt{x}$ not $+\sqrt{x}$ when $x < 0$.

$$ -1 = \sqrt{(-1)^2} = \sqrt{1} = 1$$

is an equally invalid chain of equalities becuase the square root function is not injective.

If we start at one value, and apply a function whose inverse is not injective, we can easily (by choice) end up at a different value; ie if $f^-1(x)$ is not injective, then

$$f^-1 \circ f (x) \in S$$

where S contains values other than $x$. For mathematical rigor we have to specify in which domain we are working, so that we don't simply "choose" the inverse value. If from the start we have said $x < 0$, then the inverse of $x^2$ is $-\sqrt{x^2}$ not $+\sqrt{x ^2 }$ and the fallacy would be avoided.

score 1 · Answer 10 · answered Mar 09 '14 at 02:45

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The rule $a^{b/c}=\sqrt[c]{a^b}$ is always true when $a > 0$. If $a$ is negative, it may not be true.

answered Mar 09 '14 at 02:45

TrueDefault

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score 1 · Answer 11 · answered Apr 12 '14 at 19:28

What you have done is correct but the third and so the fourth equality is not valid in mathematics. $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$

enter image description here

Here in your problem: $$\sqrt{(-1)^6}=\sqrt{(-1)^3.(-1)^3}=\sqrt{-1.-1} $$

Here since both a and b are negative

Therefore your third equality was wrong and so the fourth one.

score 0 · Answer 12 · answered Feb 16 '22 at 06:09

Your error $$(-1)^{6/2} = \sqrt{(-1)^6}\tag✗$$ can be explained in either of two ways:

in real analysis, for negative $a,$ the definition $$a^{\frac xy}:=\sqrt[y]{a^x}$$ conventionally requires $x$ and $y$ to be coprime such that $y$ is positive and odd;
for nonzero complex $a,$ the law/theorem $$a^{xy}=(a^x)^y$$ requires $x$ and $y$ to be integers.

score 0 · Answer 13 · answered Aug 22 '13 at 14:46

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The problem is in step: $$(-1)^3 = (-1)^{6/2} $$ You are squaring a number and taking root. As many people pointed out: $$ (x^2)^{1/2} $$ can be +x or -x.

answered Aug 22 '13 at 14:46

vinash85

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Dear vinash: I hate to say it, but I find this a little misleading. While it's *possible* to interpret $x^{1/2}$ in a multivalued way, it isn't really the core of the problem. I'm afraid beginners are going to get the wrong idea about the square root function, which in most basic math classes is defined to be *single valued*... – rschwieb Aug 23 '13 at 16:52
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No, in *that* step, he just rewrites the $3$ as $6/2$, which is absolutely unproblematic. It's the next step that's the problem, namely equating $(-1)^{6/2}$ with $((-1)^6)^{1/2}$. – celtschk Apr 12 '14 at 19:32

score 0 · Answer 14 · edited Aug 23 '13 at 16:51

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Think about operator associativity in $$\sqrt{(-1)^6}$$. This is constructed in following way

-1 --> -1^6 --> (-1^6)^0.5

let us solve this

(-1^6)^0.5 = 1^0.5 = 1

for more http://en.wikipedia.org/wiki/Operator_associativity

edited Aug 23 '13 at 16:51

tejas

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answered Aug 22 '13 at 19:16

Vinod Kumar

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tejas · Answer 15 · 2013-08-22T14:42:56.673

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Hi this is not a fallacy but this problem overlooks some facts i.e. there will be two square roots to every number like. $$ \sqrt{4} = +2 \text{ as well as} -2 $$ similarly $$ \sqrt{(-1)^6} = +1 \text{ or } -1 $$

So I think writing $$\sqrt{(-1)^6} = 1$$in this proof is wrong

And also people who are saying that $$(a^b)^c != a^{b*c}$$ for a <0 or something like that please calculate this in your calculator so that it proves you worng $$10^{2.5*log10(-1)} $$

edited Aug 22 '13 at 14:42

answered Aug 21 '13 at 16:25

tejas

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$\sqrt{4}=2$ **as well as $-2$** No, [that is a common misconception as well](http://math.stackexchange.com/a/294801/29335). In the context of the real numbers, $\sqrt{a}$ always denotes the nonnegative solution to $x^2-a=0$. You're right that for $a>0$ there are always two roots to $x^2-a$, but the positive one is singled out for $\sqrt{a}$. – rschwieb Aug 23 '13 at 12:40
This might actually support my point. :) Why is only +ve root considered?? – tejas Aug 23 '13 at 16:18
Because that is *how the square root function is defined*. I see above some people are being (imo overly generous) in considering $\sqrt$ as a multivalued function. Mrf's answer does not require one to mess with multivalued functions. – rschwieb Aug 23 '13 at 16:40
Before anyone gets the wrong idea, I know you *can* consider it as a multifunction, but I don't see why this discussion has to depart from basic highschool algebra like that. – rschwieb Aug 23 '13 at 16:53
Thats the points, I think its only a convention to consider $$|x|$$ as the **primary** root since the **magnitudes** are same. You dont consider a number has just 1 or 2 cube roots, the number is definitely 3, please consider the equation $$x^3-a=0$$. Similarly i think any number will have 2 definite square roots. – tejas Aug 23 '13 at 17:01
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I'm not denying you your freedom to think about multifunctions, I'm just saying that mainstream math education uses the principal roots to talk about these as single-valued functions, and so it's a little reckless and confusing to go against the mainstream convention. You are going to confuse lower level students :/ – rschwieb Aug 23 '13 at 17:07

math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

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