In Exercises 83–90, perform the indicated operation or operations. (3x+4y)^2−(3x−4y)^2

Verified step by step guidance

1. Step 1: Recognize the expression as a difference of squares. The expression is of the form (a^2 - b^2), where a = (3x + 4y) and b = (3x - 4y).
2. Step 2: Apply the difference of squares formula: a^2 - b^2 = (a + b)(a - b).
3. Step 3: Calculate (a + b) by adding the expressions: (3x + 4y) + (3x - 4y).
4. Step 4: Calculate (a - b) by subtracting the expressions: (3x + 4y) - (3x - 4y).
5. Step 5: Multiply the results from Step 3 and Step 4 to find the final expression.

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity that states that for any two terms a and b, the expression a^2 - b^2 can be factored into (a - b)(a + b). This concept is crucial for simplifying expressions that involve the subtraction of two squared terms, allowing for easier computation and understanding of the underlying structure of the equation.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, particularly those in the form (a + b)^n. The expansion can be achieved using the Binomial Theorem, which provides a formula for calculating the coefficients of the terms in the expansion. Understanding this concept is essential for accurately expanding the squared terms in the given expression.

Algebraic Manipulation

Algebraic manipulation involves the application of various algebraic rules and properties to simplify or rearrange expressions and equations. This includes operations such as factoring, distributing, and combining like terms. Mastery of algebraic manipulation is necessary to effectively perform the indicated operations in the problem, leading to a correct and simplified result.