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

Verified step by step guidance

1. Identify the expression to simplify: \(\sqrt{6}(3 + \sqrt{7})\).
2. Apply the distributive property (also known as the distributive law of multiplication over addition) to multiply \(\sqrt{6}\) by each term inside the parentheses: \(\sqrt{6} \times 3 + \sqrt{6} \times \sqrt{7}\).
3. Rewrite the multiplication of the square roots as a single square root when possible: \(3\sqrt{6} + \sqrt{6 \times 7}\).
4. Simplify the product inside the square root: \(3\sqrt{6} + \sqrt{42}\).
5. Express the final simplified form as \(3\sqrt{6} + \sqrt{42}\), noting that both terms are simplified and cannot be combined further since the radicands (6 and 42) are different.

Key Concepts

Simplifying Square Roots

Simplifying square roots involves expressing the root in its simplest form by factoring out perfect squares. For example, √6 cannot be simplified further since 6 has no perfect square factors other than 1. Understanding this helps in correctly handling radicals during operations.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property allows you to multiply a term outside the parentheses by each term inside. In this problem, √6 must be multiplied by both 3 and √7 separately before combining the results.

Multiplying Radicals

When multiplying square roots, the product rule √a * √b = √(ab) applies. This means you can multiply the numbers inside the radicals together under a single root. For example, √6 * √7 = √42, which simplifies the expression after distribution.