In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms.(5aⁿ − 7)(5aⁿ + 7)

Verified step by step guidance

1. Identify the expression as a product of the sum and difference of two terms: \((5a^n - 7)(5a^n + 7)\).
2. Recall the formula for the product of the sum and difference of two terms: \((x - y)(x + y) = x^2 - y^2\).
3. In this expression, identify \(x = 5a^n\) and \(y = 7\).
4. Apply the formula: \((5a^n)^2 - 7^2\).
5. Simplify each term: \((5a^n)^2 = 25a^{2n}\) and \(7^2 = 49\).

Key Concepts

Product of Sum and Difference

The product of the sum and difference of two terms follows the formula (a + b)(a - b) = a² - b². This identity simplifies the multiplication of expressions that are structured as a sum and a difference, allowing for a quick calculation without expanding both terms fully.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. In the given expression (5aⁿ - 7)(5aⁿ + 7), the terms 5aⁿ and 7 are combined using addition and subtraction, which are fundamental operations in algebra that allow for manipulation and simplification of expressions.

Exponent Rules

Exponent rules govern how to handle powers in algebraic expressions. In this case, 5aⁿ represents a term with a variable raised to a power, and understanding how to manipulate these terms, especially when multiplying or simplifying, is crucial for correctly applying the product of sum and difference rule.