Answer
The multiplication of two given numbers in base four is \[{{2122}_{\text{four}}}\]..
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.
\[{{32}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
Hence, first multiply \[{{3}_{\text{four}}}\] in first column with \[{{2}_{\text{four}}}\], which is above it in first column:
\[{{3}_{\text{four}}}\times {{2}_{\text{four}}}={{6}_{\text{ten}}}=\left( 1\times 4 \right)+\left( 2\times 1 \right)={{12}_{\text{four}}}\]
Now, write 2 in the first column below the horizontal line and carry 1 in the second column:
\[\overset{1}{\mathop{3}}\,{{2}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
\[{{2}_{\text{four}}}\]
Now, multiply \[{{3}_{\text{four}}}\] in first column, with \[{{3}_{\text{four}}}\], which is in the second column and add \[{{1}_{\text{four}}}\] to the product:
\[\left( {{3}_{\text{four}}}\times {{3}_{\text{four}}} \right)+{{1}_{\text{four}}}=9+1={{10}_{\text{ten}}}=\left( 2\times 4 \right)+\left( 2\times 1 \right)={{22}_{\text{four}}}\]
Write \[{{22}_{\text{four}}}\] in front of \[{{2}_{\text{four}}}\] below the horizontal line:
\[{{32}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
\[{{222}_{\text{four}}}\]
Now, repeat the whole procedure, but with \[{{2}_{\text{four}}}\]. First, place the symbol \[\times \] below \[{{2}_{\text{four}}}\] in \[{{222}_{\text{four}}}\].
\[{{32}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
\[{{222}_{\text{four}}}\]
\[\times \]
Now, multiply \[{{2}_{\text{four}}}\] in second column, with \[{{2}_{\text{four}}}\] in the first 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 \[{{0}_{\text{four}}}\] below\[{{2}_{\text{four}}}\]in \[{{222}_{\text{four}}}\] and carry 1 in the second column:
\[\overset{1}{\mathop{3}}\,{{2}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
\[{{222}_{\text{four}}}\]
\[0\times \]
Now, multiply \[{{2}_{\text{four}}}\] in the second column, with \[{{3}_{\text{four}}}\] in the second column and add \[{{1}_{\text{four}}}\] to the product:
\[\left( {{2}_{\text{four}}}\times {{3}_{\text{four}}} \right)+{{1}_{\text{four}}}=6+1={{7}_{\text{ten}}}=\left( 1\times 4 \right)+\left( 3\times 1 \right)={{13}_{\text{four}}}\].
Write\[{{13}_{\text{four}}}\] in front of \[{{0}_{\text{four}}}\]:
\[{{32}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
\[{{222}_{\text{four}}}\]
\[\underline{130\times }\]
Now, add \[{{222}_{\text{four}}}\] and \[{{130}_{\text{four}}}\] in the manner shown above:
\[{{32}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
\[{{222}_{\text{four}}}\]
\[\underline{+130\times }\]
First, write \[{{2}_{\text{four}}}\] below the symbol \[\times \] as it is:
\[{{32}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
\[{{222}_{\text{four}}}\]
\[\underline{+130\times }\]
\[{{2}_{\text{four}}}\]
Now, add \[{{2}_{\text{four}}}\] with \[{{0}_{\text{four}}}\]:
\[{{2}_{\text{four}}}+{{0}_{\text{four}}}={{2}_{\text{ten}}}=\left( 2\times 1 \right)={{2}_{\text{four}}}\]
\[{{32}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
\[{{222}_{\text{four}}}\]
\[\underline{+130\times }\]
\[{{22}_{\text{four}}}\]
Now, add \[{{2}_{\text{four}}}\] with \[{{3}_{\text{four}}}\]:
\[{{2}_{\text{four}}}+{{3}_{\text{four}}}={{5}_{\text{ten}}}=\left( 1\times 4 \right)+\left( 1\times 1 \right)={{11}_{\text{four}}}\]
\[{{32}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
\[{{222}_{\text{four}}}\]
\[\underline{+\overset{1}{\mathop{1}}\,30\times }\]
\[{{122}_{\text{four}}}\]
Now, add \[{{1}_{\text{four}}}\] with \[{{1}_{\text{four}}}\]:
\[{{1}_{\text{four}}}+{{1}_{\text{four}}}={{2}_{\text{ten}}}=\left( 2\times 1 \right)={{2}_{\text{four}}}\]
\[{{32}_{\text{four}}}\]
\[\underline{\times {{23}_{\text{four}}}}\]
\[{{222}_{\text{four}}}\]
\[\underline{+130\times }\]
\[{{2122}_{\text{four}}}\]
Now, to check whether the above obtained solution is correct, perform the multiplication by converting each number to base ten:
\[{{32}_{\text{four}}}=14\], \[{{23}_{\text{four}}}=11\] and \[{{2122}_{\text{four}}}=154\]
Since, \[14\times 11\] indeed equals 154, the solution obtained is correct.
Hence, the multiplication of two given numbers in base four is \[{{2122}_{\text{four}}}\]..