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