Question

Find the square root of 67?

Answer 1

Hint: Here in this question, we have to determine the square root of the given number, and they have mentioned that we have to determine the square root of a number by the division method. The division method is a specified method and hence we obtain the required solution for the given solution.

Complete answer:
The division method is generally explained as
Group the digits in pairs, starting with the digit in the units place.
Find the largest number whose square is equal to or just less than the first period. ...
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.
Now consider the given number of 67.
$$\begin{gathered} \begin{array}{rl}& 8.185 \\ 8|\overline{\begin{array}{rl}& \stackrel{―}{67}.\stackrel{―}{00}\stackrel{―}{00}\stackrel{―}{00}\\ & \underset{―}{64}\\ & 3\end{array}}\\ & 161|\overline{\begin{array}{rl}& 3\stackrel{―}{00}\\ & \underset{―}{161}\\ & 139\end{array}}\\ & 1628|\overline{\begin{array}{rl}& 139\stackrel{―}{00}\\ & \underset{―}{13024}\\ & 876\end{array}}\\ & 16365|\overline{\begin{array}{rl}& 876\stackrel{―}{00}\\ & \underset{―}{81825}\\ & 5775\end{array}}\end{array} \end{gathered}$$
Group the digits into pairs (for digits to the left of the decimal point, pair them from right to left) by placing a bar over it. Since our number is 67, let us represent it inside the division symbol.
Now we will consider the first two digits of $$\overline{67}.\overline{00}\overline{00}\overline{00}$$ that is 67, so we divide the number 67 by 8, when 8 is multiplied by 8 we get 64, therefore on subtracting 64 from 67 then the remainder will be 3.
Let us place a decimal point in the quotient and taking a pair of zeros for further division, so now we will have 300 this number is divided by the number 161, because when we add the number 8 twice we get 16, for the unit place we will consider the number where it’s product nearer to 300, therefore when 161 is multiplied by 1 we get 161, on subtracting the number 161 from 300 then the remainder will be 139.
Now we will take another two digits so we have 13900, we are going to divide the number 13900 by 1628. When the 1 is added to the 161 we get a 162 we choose the unit place which is nearer to 13900, therefore when 1628 is multiplied by 8 we get 13024, on subtracting the number 13024 from 13900 then the remainder will be 876.
Again, take a remaining pair of digits so we have 87600. when we add the number 1628 and 8 again we get 1636, for the unit place we will consider the number which multiples is near to 87600, therefore when 16365 is multiplied by 5 we have 81825, on subtracting the number 81825 from 87600 then the remainder will be 5775. There is no end for this division because the square root of 67 is an irrational number. So, we can stop here because we get the quotient of digit having a 3 decimal point which is enough.
Therefore, the square root of 67 is approximately equal to 8.185.

Note:
Here in this question, we should not get confused by the division method, because to determine the square root by using the division method is totally different, here we will not divide the number not only by one number. So, students should not get confused.