My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure if the number stays negative or turns into a positive. I know that if a negative number is raised to an odd power it is negative, but fractional powers are neither odd or even. Is there a general rule for dealing with these types of problems?
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$\begingroup$ Do you know what $5^\frac{2}{3}$ is? Also, do you know that a negative number does not have, for instance, a square root, and in general a negative number doesn't have an n-th root where n is even (in the real numbers)? Combining those ideas should help you work out the answer. $\endgroup$– crfMar 1, 2013 at 4:13
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3$\begingroup$ @crf Yes I do, I just do not know how the negative sign will behave when simplifying. $\endgroup$– KotMar 1, 2013 at 4:14
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5$\begingroup$ Good question! You have put your finger on a subtle issue. $\endgroup$– MJDMar 1, 2013 at 4:15
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$\begingroup$ Steven - feel free to accept answers that you find to be helpful. You can accept one answer per question (and upvote as many as you'd like). To accept an answer, click on the $\checkmark$ to the left of the answer you want to accept. (You get two reputation points for every answer accepted, too!) $\endgroup$– amWhyMar 1, 2013 at 5:45
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$\begingroup$ possible duplicate? math.stackexchange.com/q/608023/75231 $\endgroup$– CODEMar 11, 2016 at 9:17
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5 Answers
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A negative base is a point of conflict between the three commonly used meanings of exponentiation.
- For the continuous real exponentiation operator, you're not allowed to have a negative base.
- For the discrete real exponentiation operator, we allow fractional exponents with odd denominators, and $$(-a)^{b/c} = \sqrt[c]{(-a)^b}= \left( \sqrt[c]{-a} \right)^b = (-1)^b a^{b/c} $$ (and this is allowed because every real number has a unique $c$-th root)
- For the complex exponentiation operator, exponentiation is multivalued. An exponentiation with denominator $n$ generally takes on $n$ distinct values, although one is generally chosen as the "principal" value.
For $(-5)^{2/3}$, these three exponentiation operators give
- Undefined
- $\sqrt[3]{25}$
- $\omega \sqrt[3]{25}$ is the principal value. The other two are $\sqrt[3]{25}$ and $\omega^2 \sqrt[3]{25}$, where $\omega = -\frac{1}{2} + \mathbf{i} \frac{\sqrt{3}}{2}$ is a cube root of $1$.
Unfortunately, which meaning of exponentiation is meant is rarely ever stated explicitly, and has to be guessed from context.
I'm guessing that the second one is meant.
In case you're curious, here is part of the rationale for the first and third conventions.
In the first convention, 'continuity' is important. If two exponents are 'near' each other, then they should produce 'nearby' values when used to exponentiate. However, despite the fact $2/3$, $3/5$, and $\pi/5$ are all similarish in size, $(-5)^{2/3}$ and $(-5)^{3/5}$ are widely separated by the fact one 'should' be positive and the other negative. And it's not even clear that $(-5)^{\pi/5}$ should be meaningful!
For the third convention, the whole thing is like the idea of $\pm 2$ being the 'square root of 4', but for the fact the complexes cannot be cleanly separated into "negative" and "positive" to let us choose a specific one nicely.
A method is chosen for the principal value, based trying to get positive bases 'right' and trying to keep continuity as much as possible, but alas this convention gets the negative bases 'wrong'.
In some sense, this can be viewed as the principal value of $(-5)^{2/3}$ chosen to be "two-thirds of the way" from positive to negative.
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1$\begingroup$ Why do you say $\omega\sqrt[3]{25}$ is the principal value and not $\sqrt[3]{25}$? $\endgroup$– MJDMar 1, 2013 at 4:47
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2$\begingroup$ Well done. Probably TMI. Maybe it will prompt further investigation. $\endgroup$ Mar 1, 2013 at 4:47
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1$\begingroup$ @MJD: The standard choice of complex angle for a negative number is $\arg z = \pi$, and the principal $n$-th root has angle $n \pi$. $\endgroup$– user14972Mar 1, 2013 at 4:57
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4$\begingroup$ @Hurkyl: sorry for resurrecting this thread, but just for posterity, did you mean that the principal $n^\text{th}$ root has angle $\pi/n$? $\endgroup$– robjohn ♦Feb 18, 2014 at 2:51
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1$\begingroup$ @Brian: Nope; the expression I defined is only for odd denominators. (and if a fraction can't be written with an odd denominator, the exponentiation isn't defined) $\endgroup$– user14972May 17, 2014 at 15:35
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This question is several years old. Nevertheless, I'm bothered that none of the answers provided were at an introductory level. Let me then contribute an answer, which is not rigorous, but will serve as a general heuristic for students who have only recently encountered fractional exponents with real numbers.
We can interpret a base raised to a simplified$^\dagger$ fractional exponent with this heuristic: $$x^{\frac ab} = x^{\frac{\text{'power'}}{\text{'root'}}}$$ That's saying that $a$ acts like a standard integer power and $b$ acts like a standard integer root.$^{\dagger\dagger}$
Your example, $(-5)^{2/3}$, can be interpreted as squaring $-5$ and then taking the third root. Or, in the opposite order, taking the cube root of $-5$ and then squaring that result.
\begin{align} (-5)^{2/3} &= ((-5)^2)^{1/3} = \sqrt[3]{25} \approx 2.92 \\\\ \text{or}& \\\\ (-5)^{2/3} &= (\sqrt[3]{-5})^{2} \approx (-1.71)^2 \approx 2.92 \end{align}
Notice that in this particular example our base was negative. Since the denominator of the fraction was odd, we were able to solve for a real number. If the denominator were even, though, we would have no real solution, since the even root of a negative number is undefined for real numbers. Instead, we would have to turn to complex numbers for a more adequate interpretation (see the accepted answer by Hurkyl).
$^\dagger$ The fractional exponent must be simplified for the upcoming process to make sense. To see why, consider the example $(-8)^\frac 24$. What happens if you don't simplify? If you do?
$^{\dagger\dagger}$ We're assuming $a$ and $b$ are integers such that $a/b$ is a rational number. That's likely what a student has seen when first encountering fractional exponents. If $a,b$ are not integers, then the meaning is less obvious.
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8$\begingroup$ Thanks for answering 'at the level of the question', your answer is the only one that I understood. $\endgroup$– mattstSep 29, 2018 at 19:38
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$\begingroup$ In $^\dagger$ did you mean $(-8)^{2/4}$ or $(-8)^{2/6}$? $\endgroup$ Oct 22, 2021 at 7:49
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$\begingroup$ @peter.petrov I meant the current example, $(−8)^{2/4}$. If we were to follow the heuristic in the post and simplify first to $(−8)^{1/2}$, then we'd see there is no real solution, which is correct. However, if we were to follow the heuristic but forget to simplify, then we get the incorrect result $(−8)^{2/4} = 64^{1/4} \approx 2.83$. $\endgroup$– zahbazOct 23, 2021 at 18:44
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$\begingroup$ @zahbaz So does this mean that the answer is not unique? I somehow find that hard to teach myself. $\endgroup$ Nov 22, 2021 at 11:30
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$\begingroup$ @AmbicaGovind It's unique for non-negative real numbers, and there is no solution for negative real numbers. The issue $†$ only arises when improperly interpretting exponent rules. If it's a rational exponent, just simplify them first and you should be able to interpret the result correctly. See en.wikipedia.org/wiki/Exponentiation#Rational_exponents When dealing with complex numbers, the answer may not longer be unique. But complex numbers are beyond the scope of my answer. $\endgroup$– zahbazNov 29, 2021 at 21:08
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Odd roots of negative numbers are well-defined. $(-5)^{\frac 13}=-(\sqrt[3]5)$ is well defined as you can check: it is the only real number that satisfies $x^3=-5$. You can then square it to get $(-5)^{\frac 23}$ and if you square first and ask for $\sqrt[3]{25}$ you get the same result. If the denominator isn't odd you have a problem.
answered Mar 1, 2013 at 4:17
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1$\begingroup$ This is a bit confusing to me. Would I be able to factor out a negative 1 then simplify? All of these answers are a little advanced to me. This problem came up when I was working with limits to negative infinity. $\endgroup$– KotMar 1, 2013 at 4:54
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$\begingroup$ @StevenN: as long as the exponent is rational and the denominator is odd, yes. The issue is that $x^n$ is monotonic if $n$ is odd, but not if $n$ is even. This implies that $x^{\frac 1n}$ is a well defined function as long as $n$ is odd, but not if $n$ is even. This is what MJD was alluding to when s/he said this is a subtle issue. If you ask for $(-1)^{\frac 26}$ you might see something different from $(-1)^{\frac 13}$ because we don't know how (in the reals) to take a sixth root of $-1$ $\endgroup$ Mar 1, 2013 at 5:00
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The answer to your question depends on the assumed definition of exponentiation, but I think the most reasonable and consistent way to look at it is in the light of complex numbers. Thinking in polar coordinates, we have to ask what happens the magnitude and what happens to the angle. To answer that, we split the problem in the following way:
$$(-5)^{2/3}=(-1)^{2/3}*5^{2/3}$$
The term $(-1)^{2/3}$ specifies the angle and is equal to $e^{(2/3)i\pi}$, while $5^{2/3}$ specifies the real-valued magnitude. As such, the result according to how I interpret the operation is $$(-5)^{2/3} = 5^{2/3}*e^{(2/3)i\pi} = 5^{2/3}*(-\frac12+\frac{\sqrt3}{2}i)$$ which is roughly equal to $-1.462+2.532*i$.
Transformations interpreting the rational-valued exponent as a composition of exponentiation and extracting the root are problematic because of the failure of some logarithm and power identities on complex numbers.
On that note, it should be mentioned that even the transformation I used in the first equation is not generally allowed, it just works with splitting a negative real base into a positive real and a negative real, in the way I used it here.
EDIT: Pondering over the other answers, I think the reason for confusion is the assumed identity $\sqrt[a]{x}=x^{1/a}$, which isn't entirely correct. Taking the a-th root of a number is the inverse relation to taking a number to the power of a, it's supposed to return all numbers which, when taken to the power of a, become $x$. Raising $x$ to the power of the multiplicative inverse of a, on the other hand, is still just exponentiation with any real-valued exponent, where you raise the magnitude to the specified power and multiply the complex angle with the exponent, yielding one unique result.
This mostly comes down to interpretation, I propose it's just convenient to do the distinction to remove ambiguity.
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$\begingroup$ It is interesting to view as a movie the complex graphs of y=x^r for r=-infinity to r=infinity (with frame rate varying to prevent an infinite duration of the movie). $\endgroup$– amIOct 14, 2016 at 18:26
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Hint: we know that $a^\frac{b}{c}=(a^b)^\frac{1}{c}$. What conditions do you need on $a^b$ so that $(a^b)^\frac{1}{c}$ is defined when $c$ is even? What about odd? Then, what conditions can you put on $a$ and $b$ that will satisfy those last conditions?
