Simplify each radical. Assume all variables represent positive real numbers. ⁶√√5³

Verified step by step guidance

1. Rewrite the expression to clarify the radicals. The problem is the sixth root of the square root of 5 cubed, which can be written as \(\sqrt[6]{\sqrt{5^3}}\).
2. Express the inner square root as an exponent: \(\sqrt{5^3} = (5^3)^{\frac{1}{2}} = 5^{\frac{3}{2}}\).
3. Substitute this back into the sixth root: \(\sqrt[6]{5^{\frac{3}{2}}}\).
4. Use the property of radicals that \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\) to rewrite the expression as \(5^{\frac{3}{2} \times \frac{1}{6}}\).
5. Multiply the exponents: \(\frac{3}{2} \times \frac{1}{6} = \frac{3}{12} = \frac{1}{4}\), so the expression simplifies to \(5^{\frac{1}{4}}\).

Key Concepts

Radical Expressions and Simplification

Radical expressions involve roots such as square roots, cube roots, or higher-order roots. Simplifying radicals means rewriting them in their simplest form by factoring out perfect powers or combining roots when possible, making the expression easier to understand or compute.

Properties of Exponents and Roots

Exponents and roots are related through fractional exponents, where the nth root of a number can be expressed as that number raised to the 1/n power. Understanding how to manipulate these fractional exponents allows for easier simplification of nested radicals or expressions involving powers and roots.

Nested Radicals

Nested radicals occur when a radical expression contains another radical inside it. Simplifying nested radicals often involves rewriting the inner radical using exponent rules and then combining or simplifying the overall expression by applying root and exponent properties.