# Dimensions (squared, cubed, and more!!) [duplicate]

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.

## marked as duplicate by user122283, user99914, Claude Leibovici, user63181, Najib IdrissiApr 15 '14 at 6:43

• That is way too advanced for me lol. – Zachooz Apr 15 '14 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!) – colormegone Apr 15 '14 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 '14 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} \$ . – colormegone Apr 15 '14 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 '14 at 4:08

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!
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.