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

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Identify the expression to simplify: \(4\sqrt{3}(\sqrt{7} - 2\sqrt{11})\).

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 \(4\sqrt{3} \times (-2\sqrt{11})\).

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.

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 \(\sqrt{3} \times \sqrt{11} = \sqrt{33}\).

Combine the results from the previous steps to write the expression as \(4\sqrt{21} - 8\sqrt{33}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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