Simplify each radical. See Example 5.√24

Verified step by step guidance

1. Identify the prime factors of 24. The prime factorization of 24 is 2^3 * 3.
2. Rewrite the square root of 24 using its prime factors: \( \sqrt{24} = \sqrt{2^3 \times 3} \).
3. Apply the property of square roots that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
4. Separate the factors under the square root: \( \sqrt{2^3 \times 3} = \sqrt{2^2 \times 2 \times 3} = \sqrt{2^2} \times \sqrt{2 \times 3} \).
5. Simplify \( \sqrt{2^2} \) to 2, and express the remaining part: \( 2 \times \sqrt{6} \).

Key Concepts

Radical Simplification

Radical simplification involves rewriting a radical expression in a simpler form, often by factoring out perfect squares. For example, √24 can be expressed as √(4 × 6), allowing us to simplify it to 2√6, since √4 is a perfect square.

Perfect Squares

Perfect squares are numbers that can be expressed as the square of an integer. Common perfect squares include 1, 4, 9, 16, and 25. Recognizing these numbers is crucial for simplifying radicals, as they help identify factors that can be taken out of the radical.

Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors. For instance, the prime factorization of 24 is 2 × 2 × 2 × 3. This technique is useful in radical simplification, as it helps identify perfect square factors that can be simplified.