Question

Find the square root of 41 ?

Answer 1

Hint:Here in this question, we have to find the square root of the given number. There are many methods to find a square root value but here we choose a long division method to find the square root value of 41 by considering a zero’s after the decimal point and making them a pair we are going to determine.

Complete step by step answer:
The division method is generally explained as;
-Group the digits in pairs, starting with the digit in the unit's 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 41.
\[\begin{align}
 & {6.403 \\ 6}\left| \!{\overline {\,
 \begin{align}
&\overline{41}. \ \overline{00} \ \overline{00} \ \overline{00} \\
 & \underline{36} \\
 & 5 \\
\end{align} \,}} \right. \\
 & 124\left| \!{\overline {\,
 \begin{align}
  & 5\overline{00} \\
 & \underline{496} \\
 & 4 \\
\end{align} \,}} \right. \\
 & 1280\left| \!{\overline {\,
 \begin{align}
  & 4\overline{00} \\
 & \underline{0} \\
 & 400 \\
\end{align} \,}} \right. \\
& 12803\left| \!{\overline {\,
 \begin{align}
  & 400\overline{00} \\
 & \underline{38409} \\
 & 1591 \\
\end{align} \,}} \right. \\
\end{align}\]
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 41, let us represent it inside the division symbol.Now we will consider the first two digits of 41.000000 that is 41, so we divide the number 41 by 6, when 6 is multiplied by 6 we get 36, therefore on subtracting 36 from 41 then the remainder will be 5.

Let us place a decimal point in the quotient and taking a pair of zeros for further division, so now we will have 500 this number is divided by the number 124, because when we add the number 6 twice we get 12, for the unit place we will consider the number where it’s product nearer to 500, therefore when 124 is multiplied by 4 we get 496, on subtracting the number 496 from 500 then the remainder will be 4.

Now we will take another two digits so we have 400, we are going to divide the number 400 by 1280. When the 4 is added to the 124 we get a 128 we choose the unit place which is nearer to 400 we have a only option for a unit place i.e., 0, because if we take a number for a unit place from 0 to 9 it will be greater than 400 we can’t subtract it so we are taking 0 for the unit place so we have the number 1280. On multiplying 1280 by 0 we continue the further division. Any number is multiplied by zero then the product will be a zero then the reminder will be 400.

Again, take a remaining pair of digits so we have 40000. when we add the number 1280 and 0 again we get 1280, for the unit place we will consider the number which multiples is near to 40000, therefore when 12803 is multiplied by 3 we have 38409, on subtracting the number 38409 from 40000 then the remainder will be 1591. There is no end for this division because the square root of 41 is an irrational number. So, we can stop here because we get the quotient of a digit having a 3 decimal point which is enough.

Therefore, the square root of 41 is approximately equal to 6.403.

Note:Here we can’t use the prime factorization method because given number 41 is a prime number there is no any others factors for except 1 and 41 so we can‘t use prime factorization method likewise the repeated subtraction method cannot be used to find the square root of 41 because when we subtract odd number from the number 41 we will not obtain 0 as a final result, therefore we choose a long division method.