Question

How do you find the square root of \[138\]?

Answer 1

Hint: In the given question, we have been asked on how we can 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 answer:
The given number is \[138\], which is not a perfect square.
First, we find the closest square to \[138\], which is \[{12^2} = 144\].
Now, we divide \[138\] by the square root of the closest square,\[\sqrt {144} = 12\], and we get,
\[138 \div 12 = 11.5\]
Now, we find the average of \[12\] and \[11.5\], which is:
\[\dfrac{{11.5 + 12}}{2} = 11.75\]
Now, to get better approximation, we repeat the above steps:
\[138 \div 11.75 = 11.7446\]
Average: \[\dfrac{{11.75 + 11.7446}}{2} = 11.7473\]
Again, we are going to repeat:
\[138 \div 11.7473 = 11.7473\]
Hence, \[\sqrt {138} \approx 11.7473\]
This answer is accurate to four decimal places. To get more accuracy, we need to take more decimals at each step.

Note: In the given question, we had to calculate the square root of a number which is not a perfect square. We did by following the above method. The point where there is a big chance of making mistakes is when we divide the numbers. The smallest error of writing the quotient with even just one-digit wrong gives the wrong answer.