## Mathematical Powers – a Simple Illustration

Let’s consider the example of three times three. That can be written either 3 x 3 = 9, or in powers notation,**This tells us three to the second power (or three squared) equals nine. Consider the two images associated with this article. One is a square with sides equal to 3. The other is a cube with its three sides each equal to 3 as well. The former square contains within it nine 1 x 1 inch smaller squares. The cube contains within it twenty-seven 1 x 1 x 1 smaller cubes.**

3^{2} = 9

I know of no one who can draw a geometrical figure that demonstrates the mathematical powers expression 3

^{4}= 3 x 3 x 3 x 3 = 81. But we should need no such figure. The simple insight we’ve provided using the square and the cube should suffice.

## One Last Thought

For one last help, one last aid, consider this. 3^{2}= 3 x 3. And 3

^{8}consists of eight 3’s multiplied times each other, or

3^{8} = 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 6561

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Yes, we sometimes got insights into mathematical tricks from other people round us. Not that powers are mathematical tricks! But I remember being taught by the local postmaster that you could count the number of stamps in a sheet of stamps by multiplying the rows by the columns. It amazed me at the time and I ran to tell my parents about the “insight”! I have never seen stamps in cubes, though.