Simplify each radical. Assume all variables represent positive real numbers. $\(\sqrt{24m^6n^5}\)$

Verified step by step guidance

1. Start by expressing the radical \( \sqrt{24m^{6}n^{5}} \) as the square root of a product: \( \sqrt{24} \times \sqrt{m^{6}} \times \sqrt{n^{5}} \).
2. Simplify \( \sqrt{24} \) by factoring 24 into its prime factors: \( 24 = 4 \times 6 \), so \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} \). Since \( \sqrt{4} = 2 \), this becomes \( 2\sqrt{6} \).
3. Simplify \( \sqrt{m^{6}} \) by using the property \( \sqrt{a^{2k}} = a^{k} \) for positive \( a \). Here, \( m^{6} = (m^{3})^{2} \), so \( \sqrt{m^{6}} = m^{3} \).
4. Simplify \( \sqrt{n^{5}} \) by separating the exponent into an even and an odd part: \( n^{5} = n^{4} \times n^{1} = (n^{2})^{2} \times n \). Then, \( \sqrt{n^{5}} = \sqrt{(n^{2})^{2} \times n} = \sqrt{(n^{2})^{2}} \times \sqrt{n} = n^{2} \sqrt{n} \).
5. Combine all simplified parts to write the final expression as \( 2 m^{3} n^{2} \sqrt{6n} \).

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing the radicand as a product of perfect squares and other factors, then taking the square root of the perfect squares outside the radical. This process reduces the radical to its simplest form.

Properties of Exponents

Understanding how to manipulate exponents is essential, especially when variables are raised to powers inside a radical. For example, the square root of a variable to an even power can be simplified by dividing the exponent by two.

Assuming Variables are Positive

Assuming all variables represent positive real numbers allows us to simplify radicals without considering absolute value signs. This assumption ensures that the square root of a variable squared is simply the variable itself.