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Exponent Tables and Patterns

There are many interesting patterns to be found in the tables of powers of whole numbers.

One thing you may notice are the patterns in the one's digits. In the powers of $2$ table, the ones digits form the repeating pattern $2,4,8,6,2,4,8,6,\mathrm{...}$ . In the powers of $3$ table, the ones digits form the repeating pattern $3,9,7,1,3,9,7,1,\mathrm{...}$ . We leave it to you to figure out why this happens!

In the powers of $4$ table, the ones digits alternate: $4,6,4,6$ . In fact, you can see that the powers of $4$ are the same as the even powers of $2$ :

$\begin{array}{l}{4}^1=2^2\\ 4^2=2^4\\ 4^3=2^6\end{array}$ etc.

The same relationship exists between the powers of $3$ and the powers of $9$ :

The powers of $10$ are easy, because we use base $10$ : for ${10}^n$ just write a " $1$ " with $n$ zeros after it. For negative powers ${10}^{-n}$ , write " $0.$ " followed by $n-1$ zeros, and then a $1$ . The powers of $10$ are widely used in scientific notation , so it's a good idea to get comfortable with them.

Click here for more names for really big and really small numbers .

Another consequence of our use of base $10$ is a nice pattern between the negative powers of $2$ and the powers of $5$ .