The search for the square root of 52 has been a topic of much discussion and study over the years. Many mathematicians have attempted to solve this problem, but it still remains a challenge.

To begin with, let us define what a square root is. A square root of a number is a value that, when multiplied by itself, gives the original number as a result. For example, the square root of 4 is 2, because 2 multiplied by 2 equals 4.

Now, how do we go about finding the square root of 52? There are various methods, but one of the most common is the long division method. This involves dividing the number into smaller parts and then finding the square root of each part. Let us try this method.

First, we divide 52 into two parts – 5 and 2. We take the square root of 5, which is approximately 2.236, and write it down. We then multiply this value by 2, giving us 4.472. We subtract this from 52, giving us a remainder of 0.528. We bring down the next digit, which is 0, and write it next to the remainder, giving us 0.5280.

Next, we double the value of the quotient we got earlier (2), which gives us 4. We then guess a digit to add after the decimal point to make the number closer to the actual square root. In this case, we could guess 3, and we would get 4.3. We then find the square of 4.3, which is 18.49. We subtract this from 5.28, giving us a remainder of 1.79. We then bring down the next digit, which is also 0, and write it next to the remainder, giving us 1.790.

We repeat this process, doubling the quotient each time and guessing a digit after the decimal point until we get as many decimal places as we need. In this case, we could repeat it a few more times to get more decimal places, but it is clear that the square root of 52 is approximately 7.211.

There are other methods for finding the square root of a number, such as the Babylonian method and the Newton-Raphson method. The Babylonian method involves making an initial guess for the square root, then iteratively improving the guess until the desired accuracy is achieved. The Newton-Raphson method is similar but uses calculus to find the root of a function.

In conclusion, the search for the square root of 52 has been a challenging problem for mathematicians over the years. However, with the use of various methods, such as long division, the Babylonian method and the Newton-Raphson method, we can approximate the value to any degree of accuracy needed. Despite being a seemingly simple problem, the pursuit for the perfect solution continues to challenge us and push us to innovate in mathematics.