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When we talk about the 'difference of cubes,' we are dealing with an expression where one cube is subtracted from another. This concept can be written mathematically as: \[ a^{3} - b^{3} \] A cube means a number or expression raised to the power of three, like \( 125c^{6} \) which can be rewritten as \((5c^{2})^{3} \). The expression \( 216d^{3}\) can be rewritten as \((6d)^{3} \). Understanding this form helps with factoring. By recognizing the terms as perfect cubes, you can apply the factoring formula to simplify the expression.

Factoring formulas are essential tools in algebra to simplify expressions. One of the most common formulas is used for the difference of cubes: \[ a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})\] This formula breaks down the complex cubic expression into a product of simpler terms. For instance, with \( 125c^{6} - 216d^{3} \), we identify \( a = 5c^{2} \) and \( b = 6d \). Substituting these values into the formula, we get: \[ (5c^{2} - 6d)(25c^{4} + 30c^{2}d + 36d^{2})\] Using these formulas simplifies the process of factoring and helps us understand the structure of polynomials better.

Simplifying expressions means breaking them down to their simplest form. For the given expression \( 125c^{6} - 216d^{3} \), after applying the difference of cubes and factoring formulas, we need to simplify further. Let's start with expressing each term \[ (5c^{2})^{3} - (6d)^{3}\]. Applying the formula, \[ (5c^{2} - 6d)(25c^{4} + 30c^{2}d + 36d^{2})\], we'll proceed by multiplying and combining like terms if any. However, in this case, the expression is already in simplest form after factoring. Always look for further simplifications, like combining like terms or reducing fractions. Simplifying is essential to make expressions more manageable and easier to work within further calculations.