Exponent Tables and Patterns
There are many interesting patterns to be found in the tables of powers of whole numbers.
Powers of 2 |
Powers of 3 |
Powers of 4 |
| 21 = 2 | 31 = 3 | 41 = 4 |
| 22 = 4 | 32 = 9 | 42 = 16 |
| 23 = 8 | 33 = 27 | 43 = 64 |
| 24 = 16 | 34 = 81 | 44 = 256 |
| 25 = 32 | 35 = 243 | 45 = 1024 |
| 26 = 64 | 36 = 729 | 46 = 4096 |
| 27 = 128 | 37 = 2187 | 47 = 16384 |
| 28 = 256 | 38 = 6561 | 48 = 65536 |
| 29 = 512 | 39 = 19683 | 49 = 262144 |
| 210 = 1024 | 310 = 59049 | 410 = 1048576 |
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.... In the powers of 3 table, the ones digits form the repeating pattern 3, 9, 7, 1, 3, 9, 7, 1.... 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:
41 = 22
42 = 24
43 = 26
etc.
The same relationship exists between the powers of 3 and the powers of 9:
Powers of 3 |
Powers of 9 |
| 31 = 3 | 91 = 9 |
| 32 = 9 | 92 = 81 |
| 33 = 27 | 93 = 729 |
| 34 = 81 | 94 = 6561 |
| 35 = 243 | 95 = 59,049 |
| 36 = 729 | 96 = 531,441 |
| 37 = 2187 | 97 = 4,782,969 |
| 38 = 6561 | 98 = 43,046,721 |
| 39 = 19,683 | 99 = 387,420,489 |
| 310 = 59,049 | 910 = 3,486,784,401 |
The powers of 10 are easy, because we use base 10: for 10n 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.
Powers of 10 |
|
| 101 = 10 | 100 = 1 |
| 102 = 100 | 10-1 = 0.1 |
| 103 = 1000 | 10-2 = 0.01 |
| 104 = 10,000 | 10-3 = 0.001 |
105 = 100,000 (one hundred thousand) |
10-4 = 0.0001 (one ten thousandth) |
106 = 1,000,000 (one million) |
10-5 = 0.00001 (one hundred thousandth) |
107 = 10,000,000 (ten million) |
10-6 = 0.000001 (one millionth) |
108 = 100,000,000 (one hundred million) |
10-7 = 0.0000001 (one ten millionth) |
109 = 1,000,000,000 (one billion) |
10-8 = 0.00000001 (one hundred millionth) |
1010 = 10,000,000,000 (ten billion) |
10-9 = 0.000000001 (one billionth) |
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.
Powers of 2 |
Powers of 5 |
| 2-5 = 1/32 = 0.03125 | 5-5 = 1/3125 = 0.00032 |
| 2-4 = 1/16 = 0.0625 | 5-4 = 1/625 = 0.0016 |
| 2-3 = 1/8 = 0.125 | 5-3 = 1/125 = 0.008 |
| 2-2 = 1/4 = 0.25 | 5-2 = 1/25 = 0.04 |
| 2-1 = 1/2 = 0.5 | 5-1 = 1/5 = 0.2 |
| 20 = 1 | 50 = 1 |
