Final answer:
The phrase 'product of factors that are irreducible over the rationals' is synonymous with prime factorization for numbers and the factoring of polynomials into irreducible factors over the rationals. Understanding operations with roots and exponents, such as square roots and division of exponentials, is crucial for solving such problems.
Step-by-step explanation:
The term product of factors that are irreducible over the rationals refers to the expression of a number or polynomial as a product of irreducible factors, where these factors cannot be further factored into expressions with rational coefficients. This is closely related to the concept of prime factorization, which is a representation of a number as a product of prime numbers. For example, the number 12 can be factored into 2³, which is a product of the prime number 2. In terms of polynomials, a quadratic polynomial like x² - 5x + 6 can be factored over the rationals as (x - 2)(x - 3), since 2 and 3 are rational numbers and the factors are linear and cannot be further factored over the rationals.
When working with equations, it is sometimes necessary to perform operations such as taking square roots, cube roots, or higher roots to determine the final answer. Knowing how to perform these operations on your calculator is essential. For instance, to solve x² = 9, you'd need to calculate the square root of 9 to determine that x can be either 3 or -3.
To divide exponentials, as in the quotient a^n / a^m, you would subtract the exponents, resulting in a^(n-m). For example, dividing 5³ by 5² would give you 5^(3-2), which simplifies to 5. An important note is understanding fractional exponents represent roots, so x to the power of 1/2 would be represented as the square root of x, consistent with the laws of exponents.