How do you find the square root of \[41\]?

Hint:
In the given question, we have been asked how can we calculate the square root of a number which is not a perfect square, i.e., it cannot be expressed as the product of two rational numbers. But we can get to approximations by picking the closest square to it, calculating the quotient of the division between the number and the closest square’s square root. Then calculating their average and repeating the steps to get closer and closer to the approximations.

Complete step by step solution:
The given number is \[41\], which is not a perfect square.
First, we find the closest square to \[41\], which is \[{6^2} = 36\].
Now, we divide \[41\] by the square root of the closest square,\[\sqrt {36} = 6\], and we get,
\[41 \div 6 = 6.8333\]
Now, we find the average of \[6\] and \[6.8333\], which is:
\[\dfrac{{6 + 6.8333}}{2} = 6.4166\]
Now, to get better approximation, we repeat the above steps:
\[41 \div 6.4166 = 6.3896\]
Average: \[\dfrac{{6.3896 + 6.4166}}{2} = 6.4031\]
Again, we are going to repeat:
\[41 \div 6.4031 = 6.4031\]

Hence, \[\sqrt {41} \approx 6.4031\]

Note:
This question is not a perfect square, hence it is not going to have an exact answer. But, using this approximation method, we can calculate the approx. value. This answer is accurate to four decimal places. To get more accuracy, we need to take more decimals at each step.