In Exercises 1–38, multiply as indicated. If possible, simplify any radical expressions that appear in the product.(√2 + √7)²

Verified step by step guidance

1. Recognize that the expression \((\sqrt{2} + \sqrt{7})^2\) is a binomial squared, which can be expanded using the formula \((a + b)^2 = a^2 + 2ab + b^2\).
2. Identify \(a = \sqrt{2}\) and \(b = \sqrt{7}\) in the expression.
3. Calculate \(a^2 = (\sqrt{2})^2 = 2\).
4. Calculate \(b^2 = (\sqrt{7})^2 = 7\).
5. Calculate \(2ab = 2(\sqrt{2})(\sqrt{7}) = 2\sqrt{14}\).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In this context, √2 and √7 are square roots of 2 and 7, respectively. Understanding how to manipulate these expressions is crucial for simplifying and performing operations like addition, subtraction, and multiplication.

Binomial Expansion

The expression (√2 + √7)² is a binomial that can be expanded using the formula (a + b)² = a² + 2ab + b². This formula allows us to systematically calculate the square of a sum, which is essential for simplifying the expression correctly.

Simplifying Radical Expressions

After expanding the binomial, the next step is to simplify any resulting radical expressions. This involves combining like terms and reducing radicals when possible, such as simplifying √14 or recognizing that √a * √b = √(ab). Mastery of these simplification techniques is key to arriving at the final answer.