Answer
The multiplication of two given numbers in base four is, \[{{312}_{\text{four}}}\].
Work Step by Step
Since, the computation involves base four, the only digits which are allowed are 0, 1, 2 and 3.
The procedure to multiply two numbers in base four is same as in base ten.
\[{{21}_{\text{four}}}\]
\[\underline{\times {{12}_{\text{four}}}}\]
Hence, first multiply \[{{2}_{\text{four}}}\] in first column with \[{{1}_{\text{four}}}\], which is above it in first column:
\[{{2}_{\text{four}}}\times {{1}_{\text{four}}}={{2}_{\text{ten}}}=\left( 2\times 1 \right)={{2}_{\text{four}}}\]
Now, write 2 in the first column below the horizontal line:
\[{{21}_{\text{four}}}\]
\[\underline{\times {{12}_{\text{four}}}}\]
\[{{2}_{\text{four}}}\]
Now, multiply \[{{2}_{\text{four}}}\] in first column, with \[{{2}_{\text{four}}}\], which is in the second column:
\[{{2}_{\text{four}}}\times {{2}_{\text{four}}}={{4}_{\text{ten}}}=\left( 1\times 4 \right)+\left( 0\times 1 \right)={{10}_{\text{four}}}\]
Write \[{{10}_{\text{four}}}\] in front of \[{{2}_{\text{four}}}\] below the horizontal line:
\[{{21}_{\text{four}}}\]
\[\underline{\times {{12}_{\text{four}}}}\]
\[{{102}_{\text{four}}}\]
Now, repeat the whole procedure, with \[{{1}_{\text{four}}}\]. First, place the symbol \[\times \] below \[{{2}_{\text{four}}}\] in \[{{102}_{\text{four}}}\].
\[{{21}_{\text{four}}}\]
\[\underline{\times {{12}_{\text{four}}}}\]
\[{{102}_{\text{four}}}\]
\[\times \]
Now, multiply \[{{1}_{\text{four}}}\] in second column, with \[{{1}_{\text{four}}}\] in the first column:
\[{{1}_{\text{four}}}\times {{1}_{\text{four}}}={{1}_{\text{ten}}}=\left( 1\times 1 \right)={{1}_{\text{four}}}\]
Write \[{{1}_{\text{four}}}\]below\[{{0}_{\text{four}}}\] in \[{{102}_{\text{four}}}\]:
\[{{21}_{\text{four}}}\]
\[\underline{\times {{12}_{\text{four}}}}\]
\[{{102}_{\text{four}}}\]
\[1\times \]
Now, multiply \[{{1}_{\text{four}}}\] in the second column, with \[{{2}_{\text{four}}}\] in the second column:
\[{{1}_{\text{four}}}\times {{2}_{\text{four}}}={{2}_{\text{ten}}}=\left( 2\times 1 \right)={{2}_{\text{four}}}\]
Write \[{{2}_{\text{four}}}\]below\[{{1}_{\text{four}}}\] in \[{{102}_{\text{four}}}\]:
\[{{21}_{\text{four}}}\]
\[\underline{\times {{12}_{\text{four}}}}\]
\[{{102}_{\text{four}}}\]
\[\underline{21\times }\]
Now, add \[{{102}_{\text{four}}}\] and \[{{21}_{\text{four}}}\] in the manner shown above:
\[{{21}_{\text{four}}}\]
\[\underline{\times {{12}_{\text{four}}}}\]
\[{{102}_{\text{four}}}\]
\[\underline{+21\times }\]
\[{{312}_{\text{four}}}\]
Now, to check whether the above obtained solution is correct, perform the multiplication by converting each number to base ten:
\[{{21}_{\text{four}}}=9\], \[{{12}_{\text{four}}}=6\] and \[{{312}_{\text{four}}}=54\]
Since, \[9\times 6\] indeed equals 54, the solution obtained is correct.
Hence, the multiplication of two given numbers in base four is, \[{{312}_{\text{four}}}\].
