why is x^(2/2) is x and not |x| according to chat GPT

[Alan Najat]

hi, so I know that when we have sqrt(x^2) it can be simplified to |x|, but if the same thing is written like x^(2/2) it would just. become x and not |x| according to chat GPT, my question is, why and how? isn't sqrt(x^2) and x^(2/2) basically the same thing?

just to be clearer, the reason I'm asking this question is because of a question that our teacher had asked in the exam today, the question was this:
"if we have f(x)=x^(a/2) what is the value of (a) that make the question be differentiable at R-{0} ?" and the answer was a=2 and my teacher said that x^(2/2) would simply become |x| and it is differentiable at R-{0}, but then when I asked Chat GPT it said that it would become x not |x|

 

[Dr.Peterson]

Well, part of the problem is that ChatGPT can't actually think.

The bigger problem is that [imath]x^{2/2}[/imath] is ambiguous, so you can get several different answers.

• If you evaluate it as written, since [imath]2/2 = 1[/imath], [imath]x^{2/2} = x^1 = x[/imath].
• If you transform it to [imath](x^{1/2})^2[/imath], then [imath](x^{1/2})^2 = x[/imath] if [imath]x\ge0[/imath], and undefined otherwise.
• If you transform it to [imath](x^2)^{1/2}[/imath], then [imath](x^2)^{1/2}= |x|[/imath].

ChatGPT evidently took the expression literally; your teacher approached it from a different direction.

There's a lot more that could be said about this (and all the more, if you consider complex numbers); some of it is said here:

Powers of Roots and Roots of Powers – The Math Doctors

www.themathdoctors.org

That's a compilation of answers to questions on Ask Dr. Math, taking different perspectives.

 

[Cubist]

The bigger problem is that [imath]x^{2/2}[/imath] is ambiguous,

I'm not sure that I agree (therefore I'm probably wrong ). Personally, when I see [imath]x^{2/2}[/imath] then I think it's exactly notation for x^(2/2) without any ambiguity at all. But I notice that you used the word "transform" just beneath...

• If you transform it to [imath](x^{1/2})^2[/imath], then [imath](x^{1/2})^2 = x[/imath] if [imath]x\ge0[/imath], and undefined otherwise.
• If you transform it to [imath](x^2)^{1/2}[/imath], then [imath](x^2)^{1/2}= |x|[/imath].

...therefore, perhaps, you think similarly to me?

(Edit) To the OP...

I spotted that there's a constraint on the "power of power" rule that restricts when it can be applied some time ago (if using principal roots).
Basically a^(bc) doesn't always equal (a^b)^c since equivalence depends on the values of a,b and c. It seems that the teacher in the OP is implicitly using this rule in the question to break x^(2/2) into (x^2)^(1/2) when the values of a,b,c in this case don't actually allow it.

But, I have to say, that the intent of the teacher's question was fairly clear despite this. The question only makes sense if it's reinterpreted as (x^a)^(1/2). I can see that this would be very frustrating for any students who didn't see that the teacher wanted the answer a=2, but they must surely admit that any pupils who wrote this deserve some credit (despite the question not being written well/ correctly)

FYI: a^(bc) = (a^b)^c only if one of these is true...

1. a≥0 AND b,c any real
2. a,b any real AND c integer

 

[Dr.Peterson]

You've just said what I had in my mind but didn't say. The literal meaning is what ChatGPT took; the ambiguity is mostly in the mind of the reader, who might expect to do different things with it.

 

[Alan Najat]

thank you guys very much, appreciate the effort