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

Verified step by step guidance

1. Identify the expression to simplify: \(4\sqrt{3}(\sqrt{7} - 2\sqrt{11})\).
2. Apply the distributive property (also known as the distributive law of multiplication over subtraction) to multiply \(4\sqrt{3}\) by each term inside the parentheses separately: \(4\sqrt{3} \times \sqrt{7}\) and \(4\sqrt{3} \times (-2\sqrt{11})\).
3. Multiply the coefficients (numbers outside the radicals) together: for the first term, multiply 4 by 1 (since \(\sqrt{7}\) has an implied coefficient of 1), and for the second term, multiply 4 by -2.
4. Multiply the radicals by using the property \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\): for the first term, calculate \(\sqrt{3} \times \sqrt{7} = \sqrt{21}\), and for the second term, calculate \(\sqrt{3} \times \sqrt{11} = \sqrt{33}\).
5. Combine the results from the previous steps to write the expression as \(4\sqrt{21} - 8\sqrt{33}\).

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing square roots in their simplest form by factoring out perfect squares. For example, √12 can be simplified to 2√3 because 12 = 4 × 3 and √4 = 2. This process helps in making multiplication and addition of radicals easier.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by each term inside a parenthesis separately. In this problem, 4√3 is multiplied by both √7 and -2√11 individually before combining the results.

Multiplying Radicals

When multiplying radicals with the same index, multiply the numbers inside the radicals together under a single radical. For example, √a × √b = √(ab). This rule is used to multiply 4√3 by √7 and by 2√11 by combining the radicands before simplifying.