Question

Find the square root of 169.

Answer 1

Hint-In this question, we use the concept of long division method. In long division methods we have to make the remainder of the division become zero (0) and find the corresponding quotient of that division. Then the square root of any number is quotient of the division when the remainder becomes zero (0).

Complete step-by-step answer: -
 Now, we have a number 169 and we have to find the square root of 169 by using a long division method.
First we make the pair of the digits starting from the digits at one's place. For making the pair place a bar over every pair of digits. Like \[1\overline {69} \] but the left extreme has a single digit so we assume zero to make a pair with a single digit. Like \[\overline {01} \overline {69} \] , now we write as \[\overline 1 \overline {69} \].
Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the number under the extreme left bar as dividend. Divide and get the remainder.
\[ \Rightarrow 1\mathop{\left){\vphantom{1\begin{gathered}
  \overline 1 \overline {69} \\
  1 \\
  \overline {69} \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  \overline 1 \overline {69} \\
  1 \\
  \overline {69} \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, 1}\]
To the right of the remainder place the number that is under the next bar. Now double the divisor and enter it with blank on its right.
\[ \Rightarrow 1\mathop{\left){\vphantom{1\begin{gathered}
  \overline 1 \overline {69} \\
  1 \\
  \overline {2\left){\vphantom{1{69}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{69}}}} \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  \overline 1 \overline {69} \\
  1 \\
  \overline {2\left){\vphantom{1{69}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{69}}}} \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, 1}\]

Think a largest possible digit to fill the blank which will also become the new digit in the quotient such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.
\[ \Rightarrow 1\mathop{\left){\vphantom{1\begin{gathered}
  \overline 1 \overline {69} \\
  1 \\
  \overline {23\left){\vphantom{1\begin{gathered}
  69 \\
  69 \\
  \overline {00} \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  69 \\
  69 \\
  \overline {00} \\
\end{gathered} }}} \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  \overline 1 \overline {69} \\
  1 \\
  \overline {23\left){\vphantom{1\begin{gathered}
  69 \\
  69 \\
  \overline {00} \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  69 \\
  69 \\
  \overline {00} \\
\end{gathered} }}} \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {13}}\]

Now, remainder becomes zero (0) so the square root of 169 is quotient of the division.
$ \Rightarrow \sqrt {169} = 13$
So, the square root of 169 is 13.

Note- We can proceed our solution by the prime factorization of the given number. By the prime factorization we get 169 =$ 13\times 13$. Here we can write $\sqrt{169}$ = $\sqrt{{(13)}^{2}}$. So the square root of 169 is 13.