Multiply. See Example 7.√6 (3 + √2)

Verified step by step guidance

1. Identify the expression to be multiplied: \( \sqrt{6} (3 + \sqrt{2}) \).
2. Apply the distributive property: \( a(b + c) = ab + ac \).
3. Multiply \( \sqrt{6} \) by each term inside the parentheses: \( \sqrt{6} \times 3 \) and \( \sqrt{6} \times \sqrt{2} \).
4. Calculate \( \sqrt{6} \times 3 \) to get \( 3\sqrt{6} \).
5. Calculate \( \sqrt{6} \times \sqrt{2} \) to get \( \sqrt{12} \), which simplifies to \( 2\sqrt{3} \).

Key Concepts

Distributive Property

The distributive property states that a(b + c) = ab + ac. This principle allows us to multiply a single term by each term within a parenthesis. In the context of the given expression, √6 must be multiplied by both 3 and √2, which will help simplify the expression effectively.

Simplifying Radicals

Simplifying radicals involves reducing square roots to their simplest form. For example, √(a*b) can be expressed as √a * √b. In the expression √6(3 + √2), it is important to recognize how to handle the square root of 6 when multiplying with other terms, ensuring the final expression is as simplified as possible.

Combining Like Terms

Combining like terms is a fundamental algebraic skill that involves adding or subtracting terms that have the same variable or radical part. After applying the distributive property in the expression, it may result in terms that can be combined, such as constants or similar radical expressions, to achieve a more concise final answer.