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It follows from the preceding equations that = when x is an integer (this results from the repeated-multiplication definition of the exponentiation). If x is real, exp ( x ) = e x {\displaystyle \exp(x)=e^{x}} results from the definitions given in preceding sections, by using the exponential identity if x is rational, and the continuity ...
For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation. [ 2 ] [ 3 ] Thus, in the expression 1 + 2 × 3 , the multiplication is performed before addition, and the expression has the value 1 + (2 × 3) = 7 , and not (1 + 2) × 3 = 9 .
For any integer n, the last decimal digit of n 5 is the same as the last (decimal) digit of n, i.e. ()By the Abel–Ruffini theorem, there is no general algebraic formula (formula expressed in terms of radical expressions) for the solution of polynomial equations containing a fifth power of the unknown as their highest power.
An example of using Newton–Raphson method to solve numerically the equation f(x) = 0. In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign.
Cartan–Kähler theorem (partial differential equations) Cartan–Kuranishi prolongation theorem (partial differential equations) Cartan's theorem ; Cartan's theorems A and B (several complex variables) Casey's theorem (Euclidean geometry) Castelnuovo theorem (algebraic geometry) Castelnuovo–de Franchis theorem (algebraic geometry)
All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation
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