A manufacturing technician is calculating the combined volume of two different production batches, modeled by the expression ${}432c^3 + 686d^3$. After factoring out a common factor of 2, the technician obtains the simplified expression ${}2(216c^3 + 343d^3)$.

To apply the sum of cubes formula $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ to the expression inside the parentheses, which values must the technician identify as $a$ and $b$?

An inventory analyst is writing a procedure to simplify the calculation of combined storage capacities modeled by the expression ${}432c^3 + 686d^3$. Arrange the algebraic steps the analyst should follow to completely factor this expression in the correct order.

An industrial analyst is evaluating the expression ${}432c^3 + 686d^3$. True or False: This expression is considered prime and cannot be factored because the coefficients 432 and 686 are not perfect cubes.

A logistics coordinator is analyzing the expression ${}432c^3 + 686d^3$, which represents the combined volume of two different spherical storage tanks. To simplify the volume calculations, the coordinator must factor the expression. Match each component of the factoring process with its correct description.

Initial Factoring Step for Industrial Volume Expressions

An inventory analyst is factoring the expression ${}432c^3 + 686d^3$ to optimize a storage allocation model. They first factor out the greatest common factor of 2, obtaining ${}2(216c^3 + 343d^3)$. They then rewrite the sum inside the parentheses using perfect cube bases: ${}2((6c)^3 + (7d)^3)$. When applying the sum of cubes formula $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, the middle term of the trinomial factor, $-ab$, simplifies to a term of the form $k \cdot cd$, where the constant coefficient $k$ is ____.