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