Simplify each radical. Assume all variables represent positive real numbers. √24m⁶n⁵

Verified step by step guidance

1. First, break down the number 24 into its prime factors: \(24 = 2^3 \times 3\).
2. Next, express the expression under the square root as \(\sqrt{24m^6n^5} = \sqrt{2^3 \times 3 \times m^6 \times n^5}\).
3. Apply the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to separate the terms: \(\sqrt{2^3} \times \sqrt{3} \times \sqrt{m^6} \times \sqrt{n^5}\).
4. Simplify each square root: \(\sqrt{2^3} = 2\sqrt{2}\), \(\sqrt{m^6} = m^3\), and \(\sqrt{n^5} = n^2\sqrt{n}\).
5. Combine the simplified terms: \(2m^3n^2\sqrt{6n}\).

Key Concepts

Radical Simplification

Radical simplification involves reducing a radical expression to its simplest form. This process includes identifying perfect squares within the radicand (the number or expression inside the radical) and extracting them outside the radical. For example, √24 can be simplified by recognizing that 24 = 4 × 6, where 4 is a perfect square.

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying expressions involving variables. For instance, when simplifying m⁶, we can express it as (m³)², allowing us to take m³ outside the radical. This property helps in managing the powers of variables when they are part of a radical expression.

Combining Radicals

Combining radicals refers to the process of merging like terms after simplification. When simplifying expressions like √(a) + √(a), we can combine them into a single radical. This concept is essential for ensuring that the final expression is as concise as possible, which is a key goal in radical simplification.