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When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of their base. [2] Thus 3 + 5 2 = 28 and 3 × 5 2 = 75. These conventions exist to avoid notational ambiguity while allowing notation to remain brief. [4]
If n is a negative integer, is defined only if x has a multiplicative inverse. [39] In this case, the inverse of x is denoted x −1, and x n is defined as (). Exponentiation with integer exponents obeys the following laws, for x and y in the algebraic structure, and m and n integers:
By comparison, powers of two with negative exponents are fractions: for positive integer n, 2 −n is one half multiplied by itself n times. Thus the first few negative powers of 2 are 1 / 2 , 1 / 4 , 1 / 8 , 1 / 16 , etc.
Because every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, it follows that any negative fraction is less than any positive fraction. This allows, together with the above rules, to compare all possible fractions.
There is no standard notation for tetration, though Knuth's up arrow notation and the left-exponent are common. Under the definition as repeated exponentiation, n a {\displaystyle {^{n}a}} means a a ⋅ ⋅ a {\displaystyle {a^{a^{\cdot ^{\cdot ^{a}}}}}} , where n copies of a are iterated via exponentiation, right-to-left, i.e. the application ...
The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation d d x e x = e x {\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}} means that the slope of the tangent to the graph at each point is equal to its height (its y -coordinate) at that point.
Solving for , = = = = = Thus, the power rule applies for rational exponents of the form /, where is a nonzero natural number. This can be generalized to rational exponents of the form p / q {\displaystyle p/q} by applying the power rule for integer exponents using the chain rule, as shown in the next step.
The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been established for positive exponents. Also, one can readily deduce the quotient rule from the reciprocal rule and the product rule .