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In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product: r a d ( n ) = ∏ p ∣ n p prime p {\displaystyle \displaystyle \mathrm {rad} (n)=\prod _{\scriptstyle p\mid n \atop p{\text{ prime}}}p}
The nested radicals in this solution cannot in general be simplified unless the cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, we have the casus irreducibilis, in which all three real solutions are written in terms of cube roots of complex numbers. On the other hand, consider the ...
In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). [1] These operations may be performed on numbers, in which case they are often called arithmetic operations.
In mathematics, an algebraic expression is an expression build up from constants (usually, algebraic numbers) variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number powers, and roots (fractional powers).
The principal root of a positive real number is thus also a positive real number. As a function, the principal root is continuous in the whole complex plane, except along the negative real axis. An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd [2] or a radical. [3]
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b being real numbers, then its complex conjugate a − bi is also a root of P. [1]
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