Question

What is the square root of $5.5$?

Answer 1

Hint: We solve this problem by using the long division method of finding the square root of a number. We follow the steps that are used in the long division method to find the square root of 5.5. We need to calculate the square root up to 2 decimal points.

Complete step-by-step solution:
We are asked to find the square root of 5.5 which is not a perfect square.
Let us use the long division method of finding the square root of a number.
Let us follow the steps one by one.
We have the step 1 as “take the digits as pairs. If there are single digits then use 0’s wherever possible. Take the number of pairs after the decimal point to how many decimal points we need to find the square root.”
Let us find the required square root up to 2 decimal points. So, by taking the two pairs after the decimal point we get,
\[\left| \!{\overline {\,
 \overline{05}.\overline{50}\overline{00} \,}} \right. \]
We have step 2 as “take the nearest perfect square less than that of the first pair from the left side and the square root of that perfect square to the quotient. Subtract the perfect square from the first pair and borrow the second pair.”
Here, we can see that the first pair from the left side is $05$ and the nearest perfect square is 4.
So, let us use the step 2 then we get,
\[\begin{matrix}
   2 \\
   2\left| \!{\overline {\,
 \begin{align}
  & \overline{05}.\overline{50}\overline{00} \\
 & \underline{04\text{ }} \\
 & 01.50 \\
\end{align} \,}} \right. \\
\end{matrix}\]
We have the step 3 as “if you cross the decimal point then remove the decimal point from the dividend and use that decimal point in the quotient. Then double the quotient you got before and add a digit to the right of it and find the digit such that multiplying the number with the digit gives the number nearest to divisor and less than it then write that digit in the quotient and subtract the value from dividend and borrow the next pair.”
Here, we can see that the quotient is 2 and let us assume that the digit to be found as $k$ then according to the step 3 we get the equation as,
$\Rightarrow 4k\times k\le 150$
Where, $k$ is a digit.
Let us assume the value of $'k'$ from 1 until we get the value that satisfies the above equation then we get,
$\begin{align}
  & \Rightarrow 41\times 1=41 \\
 & \Rightarrow 42\times 2=84 \\
 & \Rightarrow 43\times 3=129 \\
 & \Rightarrow 44\times 4=176 \\
\end{align}$
Here, we can see that the value of $k$ is 3.
By using the step 3 to the division then we get,
\[\begin{align}
  & \text{ }2\begin{matrix}
   2.3 \\
   \left| \!{\overline {\,
 \begin{matrix}
   \overline{05}.\overline{50}\overline{00} \\
   \underline{04\text{ }} \\
\end{matrix} \,}} \right. \\
\end{matrix} \\
 & 43\left| \begin{matrix}
   0150 \\
   \underline{0129} \\
   \text{ }2100 \\
\end{matrix} \right. \\
\end{align}\]
We have step 4 as “repeat step 3 until all the pairs are completed. Then the quotient will be the required answer.”
Now, let us do step 3 again.
Here, we can see that the quotient is $23$ and let us assume that the digit to be found as $k$ then according to the step 3 we get the equation as,
$\Rightarrow 46k\times k\le 2100$
Where, $k$ is a digit.
Let us assume the value of $'k'$ from 1 until we get the value that satisfies the above equation then we get,
$\begin{align}
  & \Rightarrow 461\times 1=461 \\
 & \Rightarrow 462\times 2=924 \\
 & \Rightarrow 463\times 3=1389 \\
 & \Rightarrow 464\times 4=1856 \\
 & \Rightarrow 465\times 5=2325 \\
\end{align}$
Here, we can see that the value of $k$ is 4.
By using the step 3 to the division then we get,
\[\begin{align}
  & \text{ }2\begin{matrix}
   2.34 \\
   \left| \!{\overline {\,
 \begin{matrix}
   \overline{05}.\overline{50}\overline{00} \\
   \underline{04\text{ }} \\
\end{matrix} \,}} \right. \\
\end{matrix} \\
 & \text{ }43\left| \begin{matrix}
   \text{ }0150 \\
   \text{ }\underline{0129} \\
\end{matrix} \right. \\
 & 464\left| \begin{matrix}
   \text{ }2100 \\
   \text{ }\underline{1856\text{ }} \\
   \text{ }244 \\
\end{matrix} \right. \\
\end{align}\]
Here, we can see that the pairs are completed and the quotient is $2.34$
Therefore we can conclude that the square root of $5.5$ is $2.34$ that is,
$\therefore \sqrt{5.5}=2.34$

Note: We need to be very careful while finding the square root of non – perfect square numbers. Many mistakes can be done in the calculation parts because here we can see that we need to do some calculations that are very necessary.
Here, we can see that we decided to find the square root till 2 decimal points but we can extend that to any number of decimal points that we needed. But going on further the calculations will be a bit difficult. So, calculating square root till 2 decimal points is enough for this type of problem.