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

Verified step by step guidance

1. Recognize that the expression \((\sqrt{2x} - \sqrt{y})^2\) is a binomial squared, which can be expanded using the formula \((a - b)^2 = a^2 - 2ab + b^2\).
2. Identify \(a = \sqrt{2x}\) and \(b = \sqrt{y}\) in the expression.
3. Apply the formula: \((\sqrt{2x})^2 - 2(\sqrt{2x})(\sqrt{y}) + (\sqrt{y})^2\).
4. Simplify each term: \((\sqrt{2x})^2 = 2x\), \(-2(\sqrt{2x})(\sqrt{y}) = -2\sqrt{2xy}\), and \((\sqrt{y})^2 = y\).
5. Combine the simplified terms to get the final expression: \(2x - 2\sqrt{2xy} + y\).

Key Concepts

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, particularly those in the form of (a + b)². The formula for this expansion is (a + b)² = a² + 2ab + b². In the given expression (√2x - √y)², recognizing it as a binomial allows us to apply this formula to simplify the multiplication.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In this context, √2x and √y are radical expressions. Understanding how to manipulate these expressions, including simplifying them and combining like terms, is essential for solving problems that involve radicals, especially when they appear in products or sums.

Simplification of Expressions

Simplification of expressions involves reducing them to their simplest form, which can include combining like terms, reducing fractions, and simplifying radicals. In the context of the problem, after applying the binomial expansion, it is important to simplify any resulting radical expressions to ensure the final answer is presented in its most concise form.