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
2 1 =2 3 1 =3 4 1 =4
2 2 =4 3 2 =9 4 2 =16
2 3 =8 3 3 =27 4 3 =64
2 4 =16 3 4 =81 4 4 =256
2 5 =32 3 5 =243 4 5 =1024
2 6 =64 3 6 =729 4 6 =4096
2 7 =128 3 7 =2187 4 7 =16384
2 8 =256 3 8 =6561 4 8 =65536
2 9 =512 3 9 =19683 4 9 =262144
2 10 =1024 3 10 =59049 4 10 =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 :

4 1 = 2 2 4 2 = 2 4 4 3 = 2 6 etc.

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

Powers of 3
Powers of 9
3 1 =3 9 1 =9
3 2 =9 9 2 =81
3 3 =27 9 3 =729
3 4 =81 9 4 =6561
3 5 =243 9 5 =59,049
3 6 =729 9 6 =531,441
3 7 =2187 9 7 =4,782,969
3 8 =6561 9 8 =43,046,721
3 9 =19,683 9 9 =387,420,489
3 10 =59,049 9 10 =3,486,784,401

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 n1 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
10 1 =10 10 0 =1
10 2 =100 10 1 =0.1
10 3 =1000 10 2 =0.01
10 4 =10,000 10 3 =0.001

10 5 =100,000

(one hundred thousand)

10 4 =0.0001

(one ten thousandth)

10 6 =1,000,000

(one million)

10 5 =0.00001

(one hundred thousandth)

10 7 =10,000,000

(ten million)

10 6 =0.000001

(one millionth)

10 8 =100,000,000

(one hundred million)

10 7 =0.0000001

(one ten millionth)

10 9 =1,000,000,000

(one billion)

10 8 =0.00000001

(one hundred millionth)

10 10 =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
2 0 =1 5 0 =1