Perform the indicated operations. Assume all variables represent positive real numbers. √6(3 + √7)

Verified step by step guidance

1. Distribute the \( \sqrt{6} \) across the terms inside the parentheses.
2. Multiply \( \sqrt{6} \) by 3 to get \( 3\sqrt{6} \).
3. Multiply \( \sqrt{6} \) by \( \sqrt{7} \) to get \( \sqrt{42} \).
4. Combine the results from the distribution: \( 3\sqrt{6} + \sqrt{42} \).
5. The expression \( 3\sqrt{6} + \sqrt{42} \) is the simplified form of the original problem.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In this context, the expression √6 represents the square root of 6, which is a positive real number. Understanding how to manipulate these expressions is crucial for performing operations like addition, multiplication, and simplification.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property allows us to multiply a single term by each term within a parenthesis. In the given expression, applying the distributive property will help in multiplying √6 by both terms inside the parentheses (3 and √7), leading to a simplified result.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing them to their simplest form, which often includes combining like terms or rationalizing denominators. In this case, after applying the distributive property, it is important to simplify the resulting terms, especially when dealing with products of radicals, to ensure the final expression is as concise as possible.