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

Verified step by step guidance

1. Recognize that the given expression involves the difference of two squares: \((3x + 4y)^2 - (3x - 4y)^2\). The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\).
2. Identify \(a = (3x + 4y)\) and \(b = (3x - 4y)\) in the expression. Substitute these into the difference of squares formula.
3. Apply the formula: \((3x + 4y) - (3x - 4y)\) and \((3x + 4y) + (3x - 4y)\). Simplify each term inside the parentheses.
4. Simplify \((3x + 4y) - (3x - 4y)\) to \(8y\) and \((3x + 4y) + (3x - 4y)\) to \(6x\).
5. Multiply the simplified terms: \(8y \cdot 6x\). This gives the final simplified expression, which you can calculate if needed.

Key Concepts

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 states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This concept is essential for simplifying expressions like (3x + 4y)^2.

Difference of Squares

The difference of squares is a specific algebraic identity that states a^2 - b^2 = (a - b)(a + b). This identity is useful for factoring expressions where two squared terms are subtracted. In the given problem, recognizing that the expression can be treated as a difference of squares will simplify the calculation significantly.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. This technique is crucial in algebra for simplifying expressions and solving equations. In this case, after applying the difference of squares, factoring will help in obtaining the final simplified form of the expression.