## 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.**

3² = 9

I know of no one who can draw a geometrical figure that demonstrates the mathematical powers expression 3⁴ = 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. 3² = 3 x 3. And 3³ = 3 x 3 x 3. In the first case, the power 2 also represents the number of 3’s that are multiplied together. In the second case, the power 3 represents a similar thing. So the multiplication power expression 3⁸ consists of eight 3’s multiplied times each other, or

3⁸ = 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.