The algebraic immunity of an S-box depends on the number and type of linearly independent multivariate equations it satisfies. In this paper, techniques are developed to find the number of linearly independent, multivariate, bi-affine, and quadratic equations for S-boxes based on power mappings. These techniques can be used to prove the exact number of equations for any class of power mappings. Two algorithms to calculate the number of bi-affine and quadratic equations for any$(n,n)$S-box based on power mapping are also presented. The time complexity of both algorithms is only$O(n^2)$. To design algebraically immune S-boxes, four new classes of S-boxes that guarantee zero bi-affine equations and one class of S-boxes that guarantees zero quadratic equations are presented. The algebraic immunity of power mappings based on Kasami, Niho, Dobbertin, Gold, Welch, and inverse exponents are discussed along with other cryptographic properties and several cryptographically strong S-boxes are identified. It is conjectured that a known Kasami-like highly nonlinear power mapping is differentially$4$-uniform. Finally, an open problem to find an$(n,n)$bijective nonlinear S-box with more than$5n$quadratic equations is solved.