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Consider the following: $$x^{2/3}$$

I am having some really trivial doubts, but I would like to clarify them all, once for all. My questions are:

  • Is $x^{2/3}$ the same as $\sqrt[3]{x^2}$? And if "no", why?? This is something I've been struggling with since a while, for I got ambiguous answers.

  • Why $x^{2/3}$ has no graph in the negative $x$-axis? I mean isn't this $(x^2)^{1/3}$? then even if $x = -3$, $x^2$ makes it positive, or vice versa: $x^{1/3}$ is made positive by the successive square.

I am so stuck in this banality....

N. F. Taussig
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Heidegger
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    All your questions can probably be answered if you read your definitions of fractional exponents and of roots very carefully. If you don't have definitions that answer these questions for you, then you need to use another book. – Arthur Dec 21 '22 at 22:49
  • @N.F.Taussig You're right, I meant that. And I'm saying there is no graph because I tried with some softwares and it showed nothing. Indeed I got suspicious. – Heidegger Dec 21 '22 at 23:23

1 Answers1

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I'm guessing you asked your calculator to graph $x^{2/3}$ and it didn't show anything for negative $x$.

Non-integer powers of negative numbers are rather tricky. If the exponent is $a/b$ where $a$ and $b$ are integers and $b$ is odd, you can do it (and get a negative result if $a$ is odd, or a positive result if $a$ is even). But if $a$ is odd and $b$ is even, or if the exponent is irrational, there is no real solution (you can do it with complex numbers, but I suspect you're not yet at that level). So $x^{2/3}$ with $a=2$ even and $b=3$ odd should be ok.

However, calculators generally don't work with exact arithmetic, rather with decimal approximations. The calculator doesn't even have an exact value for $2/3$, it might use $0.666666666667$ (where the number of $6$'s depends on the calculator): that is literally $666666666667/1000000000000$. It's close to $2/3$, but not exactly the same, and it's a fraction with even denominator and odd numerator. When the calculator is asked to take a negative number to this power, it says "Oops: the exponent has an even denominator and odd numerator!" and refuses to return an answer.

Some calculators may (internally) use base $2$ rather than base $10$, but the end result is similar.

Robert Israel
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    It gets really tricky when you realize that $\frac23$ and $\frac46$ represent the exact same number, but somehow $x^{2/3}$ works for negative inputs while $x^{4/6}$ doesn't. And does $x^{0.666\ldots}$ make sense? Ultimately, in my opinion, very little is gained and in exchange sleep is lost if you venture down the path of fractional exponents with negative bases. I would personally just leave them as forbidden. – Arthur Dec 21 '22 at 23:23
  • @Arthur makes a good point worth expanding on: if $x<0$, $(x^{1/6})^4$ has the issue he mentioned, while $(x^4)^{1/6}$ doesn't, so $(x^a)^b=(x^b)^a$ breaks down. – J.G. Dec 21 '22 at 23:53
  • Indeed. The fractional exponent needs to be in reduced form in order for $(x^a)^{1/b}=(x^{1/b})^a$ to hold. – Clayton Dec 22 '22 at 00:38
  • Related, possibly helpful: https://math.stackexchange.com/questions/4412683/proofs-request-proofs-that-five-exponention-rules-hold-for-positive-real-bases/4412691#4412691 – Ethan Bolker Dec 22 '22 at 02:57