highestpower

Highestpower is a term used informally in mathematics to denote the largest exponent of a variable in a given expression, or the largest exponent of a prime dividing a number. Its meaning depends on the mathematical context.

In polynomial terminology, the highest power of a variable refers to the degree of the polynomial with respect to that variable. For a polynomial f(x) = a_n x^n + … + a_1 x + a_0 with a_n ≠ 0, the highest power of x is n, the degree in x. In multivariate polynomials, one may refer to the degree in a specific variable (the highest power of that variable) or to the total degree of the polynomial.

In number theory, the highest power of a prime p dividing an integer n is the largest power p^k such that p^k divides n. Equivalently, k is the p-adic valuation v_p(n). For example, the highest power of 2 dividing 72 is 2^3, since 2^3 divides 72 but 2^4 does not.

Computationally, determine the highest power in a polynomial by locating the term with the largest exponent of the chosen variable. For prime powers in integers, repeatedly divide by the prime until the quotient is no longer divisible; the number of divisions gives k.

Related concepts include degree, leading coefficient, exponent, and valuation. While “highestpower” is not a formal, universal term, it commonly serves as a shorthand for the highest exponent encountered in a given mathematical object.