Simplify each radical. See Example 5. √24

Verified step by step guidance

1. Identify the number inside the square root, which is 24, and look for its prime factors or perfect square factors.
2. Express 24 as a product of its factors, focusing on separating a perfect square: \(24 = 4 \times 6\).
3. Use the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to rewrite \(\sqrt{24}\) as \(\sqrt{4} \times \sqrt{6}\).
4. Simplify the square root of the perfect square: \(\sqrt{4} = 2\), so the expression becomes \(2 \times \sqrt{6}\).
5. Write the simplified form as \(2\sqrt{6}\), which is the simplified radical form of \(\sqrt{24}\).

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form by factoring the radicand into perfect squares and other factors. For example, √24 can be broken down into √(4 × 6), where 4 is a perfect square, allowing simplification to 2√6.

Perfect Squares

Perfect squares are numbers that are squares of integers, such as 1, 4, 9, 16, and 25. Recognizing perfect squares within a radicand helps simplify radicals by extracting their square roots as whole numbers, making the expression simpler and easier to work with.

Properties of Square Roots

The property √(a × b) = √a × √b allows the separation of a square root of a product into the product of square roots. This property is fundamental in simplifying radicals by breaking down complex radicands into factors that include perfect squares.