Question

How do you find the square root of 85?

Answer 1

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 \[85\], which is not a perfect square.
First, we find the closest square to \[85\], which is \[{9^2} = 81\].
Now, we divide \[85\] by the square root of the closest square,\[\sqrt {81} = 9\], and we get,
\[85 \div 9 = 9.4444\]
Now, we find the average of \[9\] and \[9.4444\], which is:
\[\dfrac{{9 + 9.4444}}{2} = 9.2222\]
Now, to get better approximation, we repeat the above steps:
\[85 \div 9.2222 = 9.2169\]
Average: \[\dfrac{{9.2222 + 9.2169}}{2} = 9.2196\]
Again, we are going to repeat:
\[85 \div 9.2196 = 9.2195\]
Average: \[\dfrac{{9.2196 + 9.2195}}{2} = 9.21955\]

Hence, \[\sqrt {85} \approx 9.21955\]

Note:
For solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we think about the concept or formula which contains the known and the unknown and pick the one which is the most suitable and the most effective for finding the answer of the given question. Then we use the results or finding of the concept and apply it to our question. It is really important to know and follow all the results of the concepts if we have to solve the question correctly, as one slightest error gives the incorrect result.