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If n is a negative integer, is defined only if x has a multiplicative inverse. [37] 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:
Φ 8 (z) = (z 8 − 1)⋅(z 4 − 1) −1 = z 4 + 1. If p is a prime number, then all the p th roots of unity except 1 are primitive p th roots. Therefore, [6] = = =. Substituting any positive integer ≥ 2 for z, this sum becomes a base z repunit. Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be ...
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 .
An integer is positive if it is greater than zero, and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: If a < b and c < d, then a + c < b + d; If a < b and 0 < c, then ac < bc
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 ...
Lowe's posted results that beat the Street's estimates, but investors are homing in on its ongoing negative sales growth. The home improvement retailer posted revenue of $20.17 billion, compared ...