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I know that when you square something you can visualize it as a 2d square. When you cube it you can visualize it as a 3d cube.

For example:

2^2 -- a 2 by 2 square

2^3 -- a 2 by 2 by 2 cube

I've been puzzling over how something would look when you make it to the power of a fraction. Is it some shape with 2D and 3D aspects? Or is it just a non-regular 3d shape.

2^(3/2)

It would be nice to have an explanation and maybe some pics :)

P.S.

I didn't really know what tags this should have so feel free to edit.

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marked as duplicate by Sanath, John, Claude Leibovici, Sami Ben Romdhane, Najib Idrissi Apr 15 at 6:43

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That is way too advanced for me lol. –  Zachooz Apr 15 at 3:16
    
You will want to read about... fractals! [en.wikipedia.org/wiki/Fractal , mathworld.wolfram.com/Fractal.html , etc.] (You can have "objects" with dimensions that are irrational numbers!) –  RecklessReckoner Apr 15 at 3:57
    
@RecklessReckoner Fractals that have fractional and irational dimension aren't really that physical dimension, they just have that property numerically by area as you increase their values. Its not a new depth that gives them that, but a lack of the normal definition of powering. The term dimension technically applies to them, but I don't think its what the asker had in mind. –  Asimov Apr 15 at 4:00
    
Yes, I know they are "embedded" in a space of the next higher integer dimension. Since Zachooz asked about how an object with fractional dimension "looked", it seems that is just what he wanted. The meaning of "dimension" had to be extended outside of its usual physical context (in fact, there are multiple definitions of dimension at this point). Naturally, we aren't talking about constructing $ \ \mathbb{R}^{3/2} \ $ . –  RecklessReckoner Apr 15 at 4:05
    
It sounds like you want to define $a^b$ for $a$ a positive real and $b$ any real. Do you accept how to define $a^b$ for $a$ a positive rational and $b$ a rational, based on the laws of exponents? This is a big step-then we pass to the reals by continuity. –  Ross Millikan Apr 15 at 4:08

2 Answers 2

Eh. I tried. I'm not an artist though.

The value $x$ is $\displaystyle 2^{3/2}$. You can square both sides to get $x^2 = 2^3$, because $(a^b)^c = a^{bc}$. This was the best pictorial description I could think of, let me know if it helped!

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so its like a rectangular prism? –  Zachooz Apr 15 at 3:28
    
Hes saying that to represent the 3/2 power, he has to represent a 3 power and a 2 power with the same value, and use both a cube and a square to show each of them. –  Asimov Apr 15 at 3:53

I hate to burst your simplified views on powers, but fractional and negative powers are just not able to be explained by spacial representation. There is a lot of math, especially involving powers and imaginary numbers in which it makes no sense, but the algebra works out, like the famous Euler Identity, which says that $e^{i\pi}+1=0$. Its completely true, tried and tested numerous ways, but it doesn't make sense at all to multiply $e$ by itself imaginary $\pi$ times. How on earth do you multiply something by itself imaginary times? Or $\pi$ times for that matter? It doesn't make sense in the traditional way, but the math works if you approach it from the right angle, so we accept it.

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Ah so it just can't be physically represented? –  Zachooz Apr 15 at 3:54
    
sadly, not really, or at least not easily. Maybe in a 3/2 dimensional universe it would be possible, but who knows if those even exist? –  Asimov Apr 15 at 3:57
    
Or at least I haven't seen a good representation of them yet. –  Asimov Apr 15 at 4:01

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