Question

Find the value of the following, using the column method: ${\left( {23} \right)^2}$

Answer 1

Hint: To find the value of ${\left( {23} \right)^2}$ using column method, Consider $a = 2\;{\text{and}}\;b = 3$ and then make three column consists of the terms present in the expansion of ${(a + b)^2}$ and then use column method to add them and finally get the result. Also check your result by finding the square of the given number through simple multiplication.

Complete step by step solution:
In order to find the value of given square number ${\left( {23} \right)^2}$ with the help of column method, we will consider $a = 2\;{\text{and}}\;b = 3$ and create three columns consisting ${a^2},\;2ab\;{\text{and}}\;{b^2}$ and put their respective values below them
\[\begin{array}{*{20}{c}}
\hline
   & {{\text{Column}}\;{\text{I}}}& & {{\text{Column}}\;{\text{II}}}& & {{\text{Column}}\;{\text{III}}} & \\
\hline
   & {{a^2}}& & {2ab}& & {{b^2}} & \\
\hline
   & 4& & {12}& & 9 &
\hline
\end{array}\]
Now underline the digit at one’s place of column three and add its ten’s place digit with column three,
\[\begin{array}{*{20}{c}}
\hline
   & {{\text{Column}}\;{\text{I}}}& & {{\text{Column}}\;{\text{II}}}& & {{\text{Column}}\;{\text{III}}} & \\
\hline
   & {{a^2}}& & {2ab}& & {{b^2}} & \\
\hline
   & 4& & {12 + 0 = 12}& & {\underline 9 } &
\hline
\end{array}\]
This time underline the digit at one’s place of column two and add its ten’s place digit with column one,
\[\begin{array}{*{20}{c}}
\hline
   & {{\text{Column}}\;{\text{I}}}& & {{\text{Column}}\;{\text{II}}}& & {{\text{Column}}\;{\text{III}}} & \\
\hline
   & {{a^2}}& & {2ab}& & {{b^2}} & \\
\hline
   & {4 + 1 = 5}& & {1\underline 2 }& & {\underline 9 } &
\hline
\end{array}\]
Now write the digit of column one followed by the underlined digit of column two then after the underlined digit of column three to get the value of the required square number $ = 529$
Therefore, ${\left( {23} \right)^2} = 529$

Note:
You can check the result by finding the value of the given square number either finding it with help of simple multiplication or with help of a calculator.
Column method is actually used in addition and subtraction in which it has some simple rules which need to be followed in order to make the addition or subtraction operation successful. This method is also used for long multiplications, in which calculations are a bit tough, this method makes the calculation easy by shrinking it to one’s and ten’s digits only.