Home

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 $2$ :

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

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

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 $n - 1$ 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 $10$

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
$2^{- 5} = \frac{1}{32} = 0.03125$	$5^{- 5} = \frac{1}{3125} = 0.00032$
$2^{- 4} = \frac{1}{16} = 0.0625$	$5^{- 4} = \frac{1}{625} = 0.0016$
$2^{- 3} = \frac{1}{8} = 0.125$	$5^{- 3} = \frac{1}{125} = 0.008$
$2^{- 2} = \frac{1}{4} = 0.25$	$5^{- 2} = \frac{1}{25} = 0.04$
$2^{- 1} = \frac{1}{2} = 0.5$	$5^{- 1} = \frac{1}{5} = 0.2$
$2^{0} = 1$	$5^{0} = 1$