This is an explanation of numerical powers, such as squares and cubes. As with my explanation of bases, there is nothing innovative here, but if you don't understand how these work then this is a good place to start.

Okay, I assume that you're familiar with basic multiplication. For instance, 2 * 3 = 6, and 8 * 9 = 72. If you compare multiplication to addition, this is much the same as comparing powers to multiplication.

When you multiply two numbers together, you are basically doing lots of addition. So, 8 * 9 = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 72.

When you take a power of a number, you are doing lots of multiplication. For instance, 5 to the power of 2 (written 5^2, or as 5 with a little 2 in the top right hand corner), means 5 multiplied by itself 2 times, i.e. 5 * 5 = 25. Similarly, 6^2 = 6 * 6 = 36. When we raise a number to the power of 2, this is referred to as squaring it.

The next stage is higher powers. For instance, if we raise something to the third power (also known as cubing it), we get 5^3 = 5 * 5 * 5 = 125.

The power can be as big as you want; for instance, you could have 5^15, which equals:

5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 = 30517578125

As you can see, the results of these calculations rise rapidly. This is known as an **exponential** increase. Here are the first 10 powers of 5:

5^1 = 5

5^2 = 25

5^3 = 125

5^4 = 625

5^5 = 3125

5^6 = 15625

5^7 = 78125

5^8 = 390625

5^9 = 1953125

5^10 = 9765625

This page was last updated on 2003-12-29 by John C. Kirk