Question

Finding square root by division method

Answer 1

Hint: We forget the decimal point and take a pair of digits from the right to left as a dividend. We find the positive integer whose square is less than the first pair of gits as the first dividend from the left and take than integer as the divisor. After division we bring down the next pair of digits with the reminder it becomes a new dividend. We double the first divisor and add a digit to the right of such that it will be less than or equal to the new dividend. We continue the division. If there are $ n $ pair of digits after the decimal point in the original number then we place the decimal point after $ n $ digits from the left. \[\]

Complete step by step answer:
We are given the number 67.24 to find the square root. We make pair of digits from the right and put bars over them as $ \overline{67}.\overline{24} $ . So our first dividend will be the number formed by the first pair of digits from the left that is 67. Our first divisor will be an integer whose square is just less than 67 which is 8 since $ {{8}^{2}}=64<7 $ . We initiate the long division.\[\]
\[\begin{gathered}
  8\mathop{\left){\vphantom{1{\overline {67} \overline {24} }}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{\overline {67} \hspace{.1 cm}\overline {24} }}}}
\limits^{\displaystyle\,\,\, 8} \\
  {\text{ }}\underline {{\text{64}}} \\
  {\text{ 3}} \\
\end{gathered} \]
So we bring down 24 from the dividend and find the new dividend with remainder as 324. We double the first divisor and get $ 8\times 2=16 $ . Now we have a place digit $ d $ on the right of 16 such that $ 16d\times d=324 $ or if that is not possible we take $ d $ such that $ 16d\times d<324 $ . We use hit and miss method find the digit as 2 since $ 162\times 2=324 $ . So we continue the long division method as;\[\]

\[\begin{gathered}
  8\mathop{\left){\vphantom{1{\overline {67} \overline {24} }}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{\overline {67} \hspace{0.1 cm}\overline {24} }}}}
\limits^{\displaystyle\,\,\, {82}} \\
  {\text{ }}\underline {{\text{64}}} \\
  {\text{162}}\left){\vphantom{1{324}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{324}}} \\
  {\hspace{0.8 cm}}\underline {{\text{324}}} \\
  {\hspace{0.8 cm}\text{ 0}} \\
  {\text{ }} \\
\end{gathered} \]
Since we have one pair of digits in 67.24 after decimal put one decimal point after one digit in 82 from left to have the square root as 8.2.\[\]

Note:
We note that the first divisors are perfect squares (integers whose square root is an integer) and we should remember perfect squares up to 20 for long division. We note that we must prioritize the digit $ d $ in the second division for equality with the new dividend rather less than equal to the new dividend.