In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product.(4√3 + 3√2) (4√3 - 3√2)

Verified step by step guidance

1. Recognize the expression as a difference of squares: \((a + b)(a - b) = a^2 - b^2\).
2. Identify \(a = 4\sqrt{3}\) and \(b = 3\sqrt{2}\).
3. Calculate \(a^2 = (4\sqrt{3})^2 = 16 \times 3 = 48\).
4. Calculate \(b^2 = (3\sqrt{2})^2 = 9 \times 2 = 18\).
5. Subtract \(b^2\) from \(a^2\) to get the result: \(a^2 - b^2 = 48 - 18\).

Key Concepts

Multiplication of Binomials

The multiplication of binomials involves applying the distributive property, often referred to as the FOIL method (First, Outside, Inside, Last). This technique helps in systematically multiplying each term in the first binomial by each term in the second binomial, ensuring that all combinations are accounted for in the resulting expression.

Radical Expressions

Radical expressions contain roots, such as square roots or cube roots. Simplifying these expressions often involves factoring out perfect squares or cubes, allowing for a clearer representation of the expression. Understanding how to manipulate and simplify radicals is crucial for solving problems that involve them.

Difference of Squares

The difference of squares is a specific algebraic identity that states that the product of two conjugates, (a + b)(a - b), equals a² - b². This identity simplifies calculations significantly, especially when dealing with radical expressions, as it allows for the direct computation of the squares of the terms involved.