What is the approximate distance between points $\left( -7,2 \right)$ and $\left( 11,-5 \right)$?

Hint: We first use the formula of distance between two points $\left( a,b \right)$ and $\left( c,d \right)$ as $d=\sqrt{{{\left( a-c \right)}^{2}}+{{\left( b-d \right)}^{2}}}$. We then convert the root value in decimal to find the approximate value of the distance.

Complete step-by-step solution:
We have to find the distance between points $\left( -7,2 \right)$ and $\left( 11,-5 \right)$.
We first find the general formula for distance between two arbitrary points.
We take two points $\left( a,b \right)$ and $\left( c,d \right)$.
The formula for distance between those two points will be $d=\sqrt{{{\left( a-c \right)}^{2}}+{{\left( b-d \right)}^{2}}}$.
For our given points $\left( -7,2 \right)$ and $\left( 11,-5 \right)$, we put the values for $a=-7,c=11$ and $b=2,d=-5$.
Therefore, the distance between those two points is $d=\sqrt{{{\left( -7-11 \right)}^{2}}+{{\left( 2+5 \right)}^{2}}}$.
Simplifying we get $d=\sqrt{{{18}^{2}}+{{7}^{2}}}=\sqrt{373}$ units.

We now try to find the decimal value of $\sqrt{373}$.
We take 2 digits as a set from the right end and complete the division. For the decimal form we take the set from the right side of the decimal.
\[\begin{align}
  & 19 \\
 & 1\left| \!{\overline {\,
 \begin{align}
  & \overline{3}\overline{73} \\
 & \underline{1} \\
\end{align} \,}} \right. \\
 & 29\left| \!{\overline {\,
 \begin{align}
  & 273 \\
 & \underline{261} \\
 & 12 \\
\end{align} \,}} \right. \\
\end{align}\]
Now we have to enter the decimal part. We keep doing the breaking in the set form till 2-digit place after decimal.
\[\begin{align}
  & 19.31 \\
 & 383\left| \!{\overline {\,
 \begin{align}
  & \overline{12}.\overline{00}\overline{00}\overline{00} \\
 & \underline{1149} \\
 & 5100 \\
\end{align} \,}} \right. \\
 & 3861\left| \!{\overline {\,
 \begin{align}
  & 5100\overline{00} \\
 & \underline{3861} \\
 & 123900 \\
\end{align} \,}} \right. \\
\end{align}\]

Note: The long-division method and arranging the set of 2 digits is different for integer and decimal. But taking double for the next division and putting a particular number is the same process for both of them. Since 373 is a non-perfect square number, we will find the value of root 373 using the long division method as shown above.