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The expression and calculation of the square root is known through Taylor series expansion, Therefore the square root of all the real positive numbers can be find by taking the reciprocal of its square root for positive numbers superior to 1. The goal of my work is to find the square root series expansion formulas directly for all the positive real numbers with explanations,

All the square root calculation methods and algorithms can not replace the square root √ function in equations, they requires an initial guess or starting value and / or many calculations steps or iterations to reach the square root at an arbitrarily precision which mean they need an algorithm to perform the calculation, until now it is impossible to deduce the square root of positive real number from a function. I have successfully build and tested many formulas to calculate the square root of all the positive real number in one step of calculation, based on a discovery of a pattern, this is not a calculation method only it is a mathematic law and formulas that illustrate the incontestable relation between the square root and the trigonometric function according to a specific pattern, where the output precision could be controlled and predetermined by determining the desired correct numbers of decimals initially that do not necessitate any guess or starting values. These formulas can replace the square root function in equations for many purposes according to a predetermined precisions.

The existing square root and cubic root calculation methods of the positive real numbers such as Newton method, continued fraction expansion method and many algorithm require an initial guess value of the square root and cubic root to reach the approximate value, this rough estimation or seed value is not always available for all real numbers, If the initial value is not close to the square root, the iterations will diverge away. My calculation methods do not necessitate an initial guess value close to the square root or cubic root, the starting value can be any positive real number less than of the number we want to calculate it's square root or cubic root, the sequence will reach always the square root or the cubic root of the number according to the methods described below. It is possible to increase the precision by increasing the number of calculations steps.