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For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the expression (+) +. These series consist of the centered triangular numbers 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166
In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So: n 4 = n × n × n × n. Fourth powers are also formed by multiplying a number by its cube.
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since p 2 k = p k , {\textstyle {\sqrt {p^{2k}}}=p^{k},} only roots of those primes having an odd power in the factorization are necessary.
A powerful number is a positive integer m such that for every prime number p dividing m, p 2 also divides m.Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a 2 b 3, where a and b are positive integers.
If a square number is represented by n points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of n; thus, square numbers are a type of figurate numbers (other examples being cube numbers and triangular numbers). In the real number system, square numbers are non-negative.
The square root is multivalued. One value can be chosen by convention as the principal value; in the case of the square root the non-negative value is the principal value, but there is no guarantee that the square root given as the principal value of the square of a number will be equal to the original number (e.g. the principal square root of ...
Steinmetz's equation, sometimes called the power equation, [1] is an empirical equation used to calculate the total power loss (core losses) per unit volume in magnetic materials when subjected to external sinusoidally varying magnetic flux.
The modular square root of can be taken this way. Having solved the associated quadratic equation we now have the variables w {\displaystyle w} and set v {\displaystyle v} = r {\displaystyle r} (if C {\displaystyle C} in the quadratic is a natural square).