1, Newton iteration method: Newton iteration method is a commonly used square root calculation method. For the number x whose square root is to be calculated, the square root can be approached step by step through iterative calculation. The specific iteration formula is as follows: [x {n+1} = frac {1} {2} left (Xn+frac {x} {xn} right)] where (xn) is the approximate square root of the nth iteration.

2. Dichotomy: Dichotomy is a method to approach the square root by gradually narrowing the interval. For a non-negative number x, its square root must be between 0 and x. We can choose an initial interval [0, x] and then keep narrowing this interval until we find a square root close enough.

3. Newton-Laveson method: Similar to Newton iteration method, Newton-Laveson method approaches the square root through iteration. The iterative formula is: [x {n+1} = frac {1} {2} left (xn+frac {x} {xn} right)]

4. Look-up table method: For some common square roots, you can make a square root look-up table in advance and directly look up the table to get the approximate value of the square root.

5. Square roots of special numbers: The square roots of some special numbers can be obtained through some simple formulas or special calculation methods, such as whether the square roots are integers or fractions.

6. Fractional expansion method: For some special square roots, it can be expressed as a fraction and then calculated. For example, [sqrt {2} =1+frac {1} {2+frac {1}}]

7. Iterative approximation method: Besides Newton iterative method and Newton Ravesen method, there are other iterative approximation methods, which can be selected according to actual needs.

8. Babylonian formula: Babylonian formula is a fast method to calculate the square root, which is especially suitable for the case of limited resources such as calculators. The Babylonian formula is in the form of [sqrt {x} = frac {1} {2} left (sqrt {x}+frac {a} {sqrt {x}} right)], where a is a constant, and a= 1 is usually selected, so that the iterative process can be faster.

9. Square root of special numbers: For some special numbers, such as complete square numbers, the square root is an integer and can be obtained directly. For example, [sqrt{9}=3]

10, classical solution: In ancient times, mathematicians used geometric methods to solve the square root. For example, for the problem that the area of a square is equal to a given value, the square root of this value can be found by constructing squares and rectangles with equal areas. This method was applied to ancient Greek mathematics.

1 1, binomial expansion method: for some special square roots, binomial theorem can be used to expand, especially for the calculation of approximate value, the finite term of expansion can be intercepted to get an approximate square root.

Pythagorean Theorem: Pythagorean Theorem provides a method to calculate the hypotenuse (square root of hypotenuse) of a right triangle. If there is a right-angled triangle, the lengths of two right-angled sides are A and B respectively, and the length of the hypotenuse is C, then there is [c = sqrt {a 2+b 2}]. This can be used to calculate the square root of some special angles.

13. Mathematical library function: In computer programming, you can directly use the mathematical library function provided by the programming language to calculate the square root. The common library function is sqrt (), which can be easily called in computer programs.

14, continued fraction expansion method: the square root can also be expressed as a continued fraction, and the approximation of the square root can be obtained by truncating the continued fraction expansion. This method has certain application in mathematical analysis.

Different computing methods are suitable for different scenarios, and choosing the appropriate method depends on the specific computing needs and the limitations of computing resources. In practical application, the most suitable square root calculation method is usually selected according to the characteristics of numerical calculation.