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This goes back to geometry:

Suppose you have a number of pebbles. What number of pebbles can you have so that you can arrange them in a square with the same number of pebbles along each side?

You could have:

  • 1 pebble
*

A square with 1 pebble along each side

  • 4 pebbles
**

**

A square with 2 pebbles along each side

  • 9 pebbles
***

***

***

A square with 3 pebbles along each side

  • 16 pebbles
****

****

****

****

A square with 4 pebbles along each side

  • 25 pebbles
*****

*****

*****

*****

*****

A square with 5 pebbles along each side

And so on. These numbers became known as square numbers(since they could be formed into a perfect square with the same number of pebbles along each side).

A question then arises: how many pebbles are needed for a square of a given side, for example 12 pebbles along each side?

With a side of 1, there is 1 row with 1 pebble in it, giving a total of 1 x 1 = 1.

With a side of 2, there are 2 rows with 2 pebbles in each, giving a total of 2 x 2 = 2

With a side of 3, there are 3 rows with 3 pebbles in each, giving a total of 3 x 3 = 9

So for a 12 sided square, 12 x 12 = 144 pebbles are needed. so you have made a square of [side] 12, or "squared 12". But 12 x 12 can also be written more compactly as 122.

So if you require a square with n on each side, there are n rows with n pebbles in each, giving a total of n x n = n2 and you are making a square of [side] n, or have "squared n".

Thus n2 is n squared; it is also n to the power 2. Thus raising a number to the power 2 is also called squaring it.

Geometry is also the reason behind "to the power 3" is also called "cubed", except in this case:

There are n layers with n rows of n pebbles in each layer, giving n x n x n = n3 and the pebbles have made a cube of [side] n or "cubed n"

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