Multiply. See Example 7. √6 (3 + √2)

Verified step by step guidance

1. Identify the expression to multiply: \(\sqrt{6} (3 + \sqrt{2})\).
2. Apply the distributive property (also known as the FOIL method for binomials) to multiply \(\sqrt{6}\) by each term inside the parentheses: \(\sqrt{6} \times 3\) and \(\sqrt{6} \times \sqrt{2}\).
3. Multiply the first term: \(\sqrt{6} \times 3 = 3\sqrt{6}\).
4. Multiply the second term: \(\sqrt{6} \times \sqrt{2} = \sqrt{6 \times 2} = \sqrt{12}\).
5. Combine the results to write the expression as \(3\sqrt{6} + \sqrt{12}\). You can then simplify \(\sqrt{12}\) further if needed by factoring it into \(\sqrt{4 \times 3}\).

Key Concepts

Multiplication of Radicals

Multiplying radicals involves applying the distributive property and simplifying the product. For example, when multiplying √a by (b + √c), multiply √a by each term inside the parentheses separately, then simplify any resulting radicals.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property is essential when multiplying expressions like √6(3 + √2), allowing you to multiply √6 by 3 and √6 by √2 separately before combining the results.

Simplification of Radicals

After multiplication, simplify radicals by factoring out perfect squares. For example, √(6*2) = √12 can be simplified to 2√3 because 12 = 4*3 and √4 = 2. Simplification makes the expression easier to interpret and use.