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**Hint:**Use the steps of the square root by long division method to find the square root of a given problem. In the question we have to find the square root of the number 186624 by division method.

First, we write the steps of doing the Long division method for finding square roots and then we solve the given question using those steps in the below.

**Complete step-by-step solution:**

First, we write the steps of doing the Long division method for finding square roots and then

we solve the given question using those steps in the below.

Step 1: First group the digits in pairs starting with the digit in the unit's place. Each pair and the remaining digit (if any) is called a period.

Step 2: Think of the largest number whose square is equal to or just less than the first period. taking this number divisor and also the quotient.

Step 3: Subtract the product of the divisor and the quotient from the first period and bring down the next period to the right of the remainder. this makes a new dividend

Step 4: Now the new divisor is obtained by taking two times the quotient and assessing the suitable digit which is also taken as the next digit of quotient, chosen in such a way that the product of the new divisor and the digits is equal or just less than the new dividend.

Step 5: Repeat steps (2) (3) (4) till all the periods have been taken up. Now the quotient so obtained is the required square root of the given number.

Now let's take the number 186624

\[\begin{align}

& \,\,\,\,\,\,\,\,\,\,\,\,\,\text{ 432} \\

& \Rightarrow 4\left| \!{\overline {\,

\begin{align}

& \overline{18}\overline{66}\overline{24} \\

& \underline{16} \\

\end{align} \,}} \right. \\

& \,\,\,\text{ 83}\left| \!{\overline {\,

\begin{align}

& 266 \\

& \underline{249} \\

\end{align} \,}} \right. \\

& \text{ 862}\left| \!{\overline {\,

\begin{align}

& 1724 \\

& \underline{1720} \\

& 0 \\

\end{align} \,}} \right. \\

\end{align}\]

**So, the square root of number 186624 is \[\text{432}\].**

**Note:**Students generally don’t know clearly how to do square root by division method, hence they should follow the steps in the solution so that the concept becomes clear. They should also do calculations clearly to avoid any errors.

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