In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms.(4x + 7y)(4x − 7y)

Verified step by step guidance

1. Identify the expression as a product of the sum and difference of two terms: \((a + b)(a - b)\).
2. Recall the formula for the product of the sum and difference: \((a + b)(a - b) = a^2 - b^2\).
3. In the given expression \((4x + 7y)(4x - 7y)\), identify \(a = 4x\) and \(b = 7y\).
4. Apply the formula: \((4x)^2 - (7y)^2\).
5. Simplify each term: \((4x)^2 = 16x^2\) and \((7y)^2 = 49y^2\), resulting in \(16x^2 - 49y^2\).

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 by eliminating the middle terms, resulting in a difference of squares. In the given expression, 4x and 7y are the two terms, allowing us to apply this rule directly.

Difference of Squares

The difference of squares is a specific algebraic identity that states that the product of two conjugates results in the square of the first term minus the square of the second term. This concept is crucial for simplifying expressions like (4x + 7y)(4x - 7y), as it leads to a straightforward calculation of (4x)² - (7y)², which is 16x² - 49y².

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Understanding how to manipulate these expressions, including addition, subtraction, multiplication, and applying identities, is fundamental in algebra. In this case, recognizing the structure of the expression (4x + 7y)(4x - 7y) as a product of two binomials is essential for applying the appropriate multiplication rule.