I need help understanding this. Write each of the following base three numerals in expanded notation.

  1. $22_3$
  2. $212_3$
  3. $12110_3$
  • 3
    $\begingroup$ What is expanded notation? My guess is something like $123_{10}=1\cdot 10^2+2\cdot 10^1 +3\cdot 10^0$. If so, what is your problem? $\endgroup$ Mar 12, 2015 at 3:34
  • $\begingroup$ I don't understand what the base three means and expanded notation. $\endgroup$
    – Alli
    Mar 12, 2015 at 3:38
  • $\begingroup$ We don't have a single symbol for every number, so we repeat symbols and use the location of the digit to give us more information. Where in decimal (our usual base 10), the number $234 = 200+30+4=2\cdot 10^2 + 3\cdot 10^1 + 4\cdot 10^0$, the location tells us "how much that digit is worth (in terms of powers of ten)". In a different base, say base $b$, you have $112_b=1\cdot b^2 + 1\cdot b^1 + 2\cdot b^0$ and the location of the digit tells us "how much that digit is worth (in terms of powers of b). So, for base 3, $22_3 = ...$ and $212_3=...$ $\endgroup$
    – JMoravitz
    Mar 12, 2015 at 4:08
  • 1
    $\begingroup$ "Expanded notation" is not a term I recognize, so maybe it is defined in your text (if you have one). I took a guess, which I thought was a reasonable one. What do you know about base 3? You should be able to make an analogy from my base 10 answer to base 3(Hint: replace 10s by 3s, but that won't show you understand what base 3 means) $\endgroup$ Mar 12, 2015 at 4:18

2 Answers 2


Surely this means to expand them into base ten ...

So 22 in base three is actually 8 in base ten (this is 2 + (2*3)). 212 in base three is actually 14 in base ten (this is 2 + 1*3 + 2*9). 12110 in base three is actually 147 in base ten.


If you are being asked to write numbers in the expanded notation for Base $3$ such as $112201_3$ (just a random number I made up) then you would need to write it like so (starting from the $3^5$th place, the 5th place of the digit)

$$112201_3 = 1.3^5 + 1.3^4 + 2.3^3 + 2.3^2+ 0.3^1 + 1.3^0$$

Use this video to help yourself understand https://www.youtube.com/watch?v=vOyiHMa-mtQ


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