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 $2,4,8,6,2,4,8,6,\mathrm{...}$ . In the powers of $3$ table, the ones digits form the repeating pattern $3,9,7,1,3,9,7,1,\mathrm{...}$ . 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
$9$

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


${10}^{1}=10$  ${10}^{0}=1$ 
${10}^{2}=100$  ${10}^{1}=0.1$ 
${10}^{3}=1000$  ${10}^{2}=0.01$ 
${10}^{4}=\mathrm{10,000}$  ${10}^{3}=0.001$ 
${10}^{5}=\mathrm{100,000}$ (one hundred thousand) 
${10}^{4}=0.0001$ (one ten thousandth) 
${10}^{6}=\mathrm{1,000,000}$ (one million) 
${10}^{5}=0.00001$ (one hundred thousandth) 
${10}^{7}=\mathrm{10,000,000}$ (ten million) 
${10}^{6}=0.000001$ (one millionth) 
${10}^{8}=\mathrm{100,000,000}$ (one hundred million) 
${10}^{7}=0.0000001$ (one ten millionth) 
${10}^{9}=\mathrm{1,000,000,000}$ (one billion) 
${10}^{8}=0.00000001$ (one hundred millionth) 
${10}^{10}=\mathrm{10,000,000,000}$ (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

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