Question

Find the value of root 5 up to 3 decimal places.

Answer 1

Hint: In this type of question students have to use the concept of long division method to find the square of 5. We know that in the long division method each pair of numbers in the dividend will give one digit of the square root. In this method we follow five major steps that are divide, multiply, subtract, bring down and repeat.

Complete step by step answer:
Now we have to find the square root of 5 up to 3 decimal places.
First, we will group the digits in pairs starting from the unit place.
Now we will rewrite \[5\] as \[5.000000\] since each pair of the numbers in the dividend will give one digit of the square root.
Pairing the numbers starting from the unit place we get, \[\overline{5}.\overline{00}\overline{00}\overline{00}\]
Now, we have to find the number, which when multiplied by itself, is less than or equal to the first pair.
Here, as the first pair is \[5\], the number will be 2.
\[\begin{align}
  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2 \\
 & \Rightarrow 2\left| \!{\overline {\,
 \begin{align}
  & \overline{5}.\overline{00}\overline{00}\overline{00} \\
 & 4 \\
 & \overline{100\text{ }} \\
\end{align} \,}} \right. \\
\end{align}\]
Now we will bring down the next pair \[00\] which will give us the next dividend is \[100\].
Let us add \[\text{2}\] and \[\text{2}\] to get \[4\]. Now let this \[4\] at ten’s place and find a number at unit place such that when multiplied by the same number the number become less than or equal to new dividend that is \[100\]
Thus we get the number \[42\], as we have \[42\times 2=84\]
\[\begin{align}
  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2.2 \\
 & \Rightarrow 2\left| \!{\overline {\,
 \begin{align}
  & \overline{5}.\overline{00}\overline{00}\overline{00} \\
 & 4 \\
\end{align} \,}} \right. \\
 & \,\,\,\text{ 42}\left| \!{\overline {\,
 \begin{align}
  & 100\text{ } \\
 & \text{84} \\
 & \overline{1600\text{ }} \\
\end{align} \,}} \right. \\
\end{align}\]
Now by bringing down the next pair \[00\] will give us the next dividend \[1600\].
Now add \[42\] and \[2\] which will give us \[44\].
Now we have to find a number with 4 at hundreds and tens place when multiplied by a number at unit place will give us the number which is less than or equal to new dividend
Thus we have, \[443\times 3=1329\]
\[\begin{align}
  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2.23 \\
 & \Rightarrow 2\left| \!{\overline {\,
 \begin{align}
  & \overline{5}.\overline{00}\overline{00}\overline{00} \\
 & 4 \\
\end{align} \,}} \right. \\
 & \,\,\,\text{ 42}\left| \!{\overline {\,
 \begin{align}
  & 100\text{ } \\
 & \text{84} \\
\end{align} \,}} \right. \\
 & \text{443 }\left| \!{\overline {\,
 \begin{align}
  & 1600\text{ } \\
 & \text{1329} \\
 & \overline{27100\text{ }} \\
\end{align} \,}} \right. \\
\end{align}\]
By continuing the long division process we get,
\[\begin{align}
  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2.236 \\
 & \Rightarrow 2\left| \!{\overline {\,
 \begin{align}
  & \overline{5}.\overline{00}\overline{00}\overline{00} \\
 & 4 \\
\end{align} \,}} \right. \\
 & \,\,\,\,\text{ 42}\left| \!{\overline {\,
 \begin{align}
  & 100\text{ } \\
 & \text{84} \\
\end{align} \,}} \right. \\
 & \,\text{443 }\left| \!{\overline {\,
 \begin{align}
  & 1600\text{ } \\
 & \text{1329} \\
\end{align} \,}} \right. \\
 & 4466\left| \!{\overline {\,
 \begin{align}
  & 27100 \\
 & 26796 \\
 & \overline{304\text{ }} \\
\end{align} \,}} \right. \\
\end{align}\]
Hence, the square root of \[5\] up to three decimal places is \[2.236\]

Note: In this type of question students have to follow each and every step of the long division process carefully. Students have to remember to add the last digit of the divisor to the present divisor to obtain a new divisor. Also students have to remember that at every step they have to find a digit for the unit place of the divisor such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than the dividend.