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The expression \(27x^{3n} + y^{3n}\) is an example of what we call a "sum of cubes." In algebra, a cube refers to a number or an expression raised to the power of three. The sum of cubes is a specific form of binomial, which is given as \(a^3 + b^3\). In our expression, if we choose \(a = 3x^n\) and \(b = y^n\), then it perfectly fits this format. Recognizing forms like sum of cubes is very helpful, as it allows us to use special factorization formulas. These help simplify or break down complex expressions into more manageable parts. Without recognizing the sum of cubes, we would struggle to simplify expressions efficiently.

An algebraic expression is a combination of numbers, variables, and operations (such as addition, subtraction, multiplication, and division) that represents a particular value. In the original problem, the expression \(27x^{3n} + y^{3n}\) is an algebraic expression. Here, 27 and \(y\) are coefficients, \(x^{3n}\) and \(y^{3n}\) are the terms, and \(+\) is the operation connecting them.

• Variables: Letters or symbols that stand in for unknown or changeable values, like \(x\) and \(y\) in this expression.
• Coefficients: Numbers that are multiplied by the variables, such as 27 and 1 (implied in front of \(y^{3n}\)).
• Powers/Exponents: In the expression, \(3n\) dictates how many times \(x\) and \(y\) are multiplied by themselves.

Understanding the components of algebraic expressions is essential. It allows you to manipulate them according to the rules of algebra, facilitating simplification and solving of equations.

Simplifying expressions means rewriting them in their easiest-to-understand form without changing their value. With the expression \(27x^{3n} + y^{3n}\), we have already identified it as a sum of cubes. We can simplify it by applying a specific factorization formula.The formula for the sum of cubes is \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Using our identified \(a = 3x^n\) and \(b = y^n\), we substitute into this formula:

• \((3x^n + y^n)\)
• \((9x^{2n} - 3x^n y^n + y^{2n})\)

Combining these, we achieve a simplified factored form of the original expression: \[(3x^n + y^n)(9x^{2n} - 3x^n y^n + y^{2n})\] By transforming the expression using the sum of cubes formula, we've simplified it. This makes it easier to understand and work with in further algebraic manipulations or evaluations.