Perform the indicated operations. Assume all variables represent positive real numbers. 4√3(√7 - 2√11)

Verified step by step guidance

1. Distribute the term \(4\sqrt{3}\) to each term inside the parentheses.
2. Multiply \(4\sqrt{3}\) by \(\sqrt{7}\) to get \(4\sqrt{3} \times \sqrt{7} = 4\sqrt{21}\).
3. Multiply \(4\sqrt{3}\) by \(-2\sqrt{11}\) to get \(-8\sqrt{33}\).
4. Combine the results from the distribution: \(4\sqrt{21} - 8\sqrt{33}\).
5. The expression simplifies to \(4\sqrt{21} - 8\sqrt{33}\).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this context, the expression includes square roots of numbers, which can be simplified or manipulated according to algebraic rules. Understanding how to simplify radical expressions is crucial for performing operations like addition, subtraction, or multiplication.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing us to multiply a single term by each term within a parenthesis. In the given expression, applying the distributive property is essential to correctly multiply 4√3 by each term in the expression (√7 - 2√11), ensuring all parts are accounted for in the final result.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable and exponent. In the context of radical expressions, this means simplifying the results of operations to group terms with the same radical part. This step is important for presenting the final answer in its simplest form, making it easier to interpret.