How do you simplify \[\sqrt {65} \]?

Hint: In the given question, we have been asked on 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 answer:
The given number is \[65\], which is not a perfect square.
First, we find the closest square to \[65\], which is \[{8^2} = 64\].
Now, we divide \[65\] by the square root of the closest square,\[\sqrt {64} = 8\], and we get,
\[65 \div 8 = 8.125\]
Now, we find the average of \[8\] and \[8.125\], which is:
\[\dfrac{{8 + 8.125}}{2} = 8.0625\]
Now, to get better approximation, we repeat the above steps:
\[65 \div 8.0625 = 8.0620\]
Average: \[\dfrac{{8.0625 + 8.0620}}{2} = 8.0622\]
Again, we are going to repeat:
\[85 \div 8.0622 = 8.0622\]
Hence, \[\sqrt {65} \approx 8.0622\],
This answer is accurate to four decimal places. To get more accuracy, we need to take more decimals at each step.

Note: So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. It is very important that we know the steps, as it depends on the steps that how our answer is going to take a turn, whether it is going to be right or is it going to be wrong. We get the steps right, and everything will be just as we want.