Dimensions (squared, cubed, and more!!)

Question

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.

Answer 1

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!

Answer 2

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.